[{"data":1,"prerenderedAt":1267},["ShallowReactive",2],{"layer:lines:investigate":3},{"layer":4,"contentHash":1246,"dependencyHashes":1247,"approval":1260,"releaseId":1266},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1241,"reviewStatus":1242,"authoring":1243},1,"lines","en","investigate","Test it: predictions about points and lines","Count, fold, measure and hunt for counterexamples","Predict and count how many lines, segments, rays and crossing points some points and lines can make; run a measuring experiment; beat optical illusions; and sort claims into always, sometimes and never true.",[13,14,15,16,17],"Count lines through sets of points and segments on a line, and find the n × (n − 1) ÷ 2 pattern.","Work out which numbers of crossing points are possible for three and four lines.","Compare measuring methods by experiment and explain parallax error.","Use counterexamples to decide whether statements are always, sometimes or never true.","Use paper folding to make perpendicular and parallel lines and test concurrency.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Names and definitions (Understand)",{"label":32,"value":33},"Chapters","12",{"label":35,"value":36},"Labs","Count match, two sorts, line spotter",{"label":38,"value":39},"You need","Ruler, divider, scrap paper, a friend",[41,45,74,80,114,119,125,139,142,162,177,181,186,189,212,215,227,258,268,279,282,294,299,302,311,320,331,336,348,351,368,423,428,431,459,463,474,479,482,493,529,534,539,542,607,629,634,637,657,668,673,679,684,687,696,699,722,731,743,748,751,763,792,817,829,834,837,869,873,882,887,890,899,908,919,928,932,941,952,956,1011,1024,1029,1064,1097,1201,1205,1219,1224,1229],{"id":42,"type":43,"markdown":44},"intro-investigate","prose","Mathematicians rarely start with a rule. They start with a **question**, try some examples, notice a pattern, make a guess (a **conjecture**), and then test it hard, hunting for a case that breaks it (a **counterexample**).\n\nIn this layer you will do exactly that with points and lines. How many lines can four points make? How many segments hide on a line with ten points marked on it? Can three lines cross in exactly two places? Does measuring with a divider really beat measuring with a ruler? You will need a pencil, a ruler, some scrap paper and, for one experiment, a friend.",{"id":46,"type":47,"title":48,"items":49},"steps-method","steps","The investigator's loop",[50,54,58,62,66,70],{"title":51,"tag":52,"text":53},"Ask","a clear question","\"How many segments are there on a line with 5 marked points?\"",{"title":55,"tag":56,"text":57},"Predict","commit first","Write down a guess before you count. Being wrong is useful.",{"title":59,"tag":60,"text":61},"Try small cases","1, 2, 3, 4…","Draw and count carefully. Make a table.",{"title":63,"tag":64,"text":65},"Spot a pattern","look at the differences","How does each answer grow from the one before?",{"title":67,"tag":68,"text":69},"Test","hunt for a counterexample","Try a case you have not drawn yet. Does the pattern still hold?",{"title":71,"tag":72,"text":73},"Explain","why must it be so?","A reason turns a pattern into something you can trust. Deepen does this properly.",{"id":75,"type":76,"variant":77,"title":78,"markdown":79},"careful-pattern","callout","careful","Patterns can lie","A pattern that works for the first few cases is a **conjecture**, not a fact. Mark 1, 2, 3, 4 and 5 points on a circle and join every pair: the circle is cut into 1, 2, 4, 8 and 16 pieces. It looks like doubling, so 6 points should give 32. Try it carefully: the most you can get is **31**. Only an explanation, not a pattern, makes a result certain.",{"id":81,"type":82,"caption":83,"columns":84,"rows":89},"table-circle-trap","table","The circle trap: join every pair of n points on a circle (points placed so that no three chords meet inside)",[85,86,87,88],"Points on the circle","Chords (pairs)","Pieces of the circle","Doubling guess",[90,93,95,98,101,105,109],[91,92,91,91],"1","0",[94,91,94,94],"2",[96,96,97,97],"3","4",[97,99,100,100],"6","8",[102,103,104,104],"5","10","16",[99,106,107,108],"15","31","32",[110,111,112,113],"7","21","57","64",{"id":115,"type":76,"variant":116,"title":117,"markdown":118},"obs-circle","observation","Where the doubling breaks","The pieces follow 1, 2, 4, 8, 16 and then 31, 57, not 32, 64. The real rule counts chords and their crossing points inside the circle: pieces = 1 + (number of chords) + (number of crossings). With 6 points there are 15 chords and at most 15 crossings, giving 1 + 15 + 15 = 31. A pattern of five matching cases still failed at the sixth.",{"id":120,"type":121,"title":122,"eyebrow":123,"navLabel":124},"ch01","chapter","How many lines can some points make?","Chapter 01","1 Points and lines",{"id":126,"type":127,"prompt":128,"options":129,"explanation":138},"predict-four","prediction","Mark four points on paper, with no three of them on one line. Join every pair with a line. How many different lines do you get?",[130,132,134,136],{"id":131,"label":97},"a",{"id":133,"label":99},"b",{"id":135,"label":100},"c",{"id":137,"label":33},"d","**6.** Call the points A, B, C and D. The pairs are AB, AC, AD, BC, BD and CD. Because no three points are on one line, each pair gives its own line. Many people guess 4 (one for each point) or 8; drawing it carefully settles it.",{"id":140,"type":43,"markdown":141},"four-cases","Now allow some points to line up. With four points, three different things can happen:\n\n- **All four collinear:** every pair gives the same line, so there is **1** line.\n- **Exactly three collinear:** those three give 1 line between them, and the fourth point joins each of the other three with its own line: 1 + 3 = **4** lines.\n- **No three collinear:** all 6 pairs give different lines: **6** lines.\n\nHere is the surprise: 2, 3 and 5 are **impossible** for four points. A careful computer search over every way of placing four points on a small grid finds only the answers 1, 4 and 6.",{"id":143,"type":82,"caption":144,"columns":145,"rows":150},"table-points-lines","Number of different lines through pairs of points",[146,147,148,149],"Points","All collinear","Possible answers","Most (no three collinear)",[151,155,156,158,160],[91,152,153,154],"not applicable","endlessly many lines through it, but no pairs","0 pairs",[94,91,91,91],[96,91,157,96],"1 or 3",[97,91,159,99],"1, 4 or 6",[102,91,161,103],"1, 5, 6, 8 or 10",{"id":163,"type":164,"itemId":165,"prompt":166,"check":167,"hints":171,"feedback":174},"prac-five-lines","practice","lines.investigate-five-lines","Five points, no three of them on one line. How many different lines join pairs of them?",{"kind":168,"answer":169,"tolerance":170},"number",10,0,[172,173],"List the pairs: AB, AC, AD, AE, then BC…","Each of the 5 points joins 4 others, but that counts every line twice.",{"correct":175,"incorrect":176},"Yes: 5 × 4 ÷ 2 = 10.","Each point joins 4 others: 5 × 4 = 20, but each line was counted from both ends, so 20 ÷ 2 = **10**.",{"id":178,"type":76,"variant":116,"title":179,"markdown":180},"obs-three-collinear","Why a collinear triple costs two lines","When three points A, B and C are collinear, the three pairs AB, BC and AC give **one** line instead of three, so you lose 2 lines. With five points and exactly one collinear triple you get 10 − 2 = 8 lines. That is why 9 never appears: losing lines always happens in bigger jumps.",{"id":182,"type":121,"title":183,"eyebrow":184,"navLabel":185},"ch02","Counting segments on a line","Chapter 02","2 Counting segments",{"id":187,"type":43,"markdown":188},"seg-count-intro","Mark some points on a straight line. Every **pair** of marked points is the two ends of a segment. How many segments are there?\n\nStart small. With 2 points A and B there is 1 segment. Add a third point C: now there are AB, BC and AC, which is 3. Keep going and record the results.",{"id":190,"type":82,"caption":191,"columns":192,"rows":196},"table-seg-count","Segments on a line with n marked points",[193,194,195],"Marked points","Segments","Increase from the row above",[197,199,201,203,205,207,209],[94,91,198],"start",[96,96,200],"+2",[97,99,202],"+3",[102,103,204],"+4",[99,106,206],"+5",[110,111,208],"+6",[100,210,211],"28","+7",{"id":213,"type":43,"markdown":214},"seg-pattern","Look at the last column: +2, +3, +4, +5… Each new point makes a new segment with **every point already there**. The 5th point joins 4 old points, so it adds 4 segments.\n\nThe totals 1, 3, 6, 10, 15, 21, 28 are the **triangular numbers**, the same numbers you get stacking bowling pins or oranges in a triangle. There is a shortcut: each of the n points is the end of (n − 1) segments, giving n × (n − 1) ends, and every segment has 2 ends, so\n\n**segments = n × (n − 1) ÷ 2**.",{"id":216,"type":127,"prompt":217,"options":218,"explanation":226},"predict-ten-points","A line has **10** points marked on it. How many segments do they make?",[219,220,222,224],{"id":131,"label":103},{"id":133,"label":221},"20",{"id":135,"label":223},"45",{"id":137,"label":225},"90","**45.** Using the shortcut: 10 × 9 ÷ 2 = 45. If you guessed 90, you counted every segment twice, once from each end.",{"id":228,"type":229,"component":230,"componentVersion":5,"config":231,"objective":256,"textAlternative":257},"lab-match-counts","interactive","match-pairs",{"prompt":232,"mode":233,"pairs":234},"Match the number of points on a line to the number of segments they make.","connect",[235,238,241,244,247,250,253],{"a":236,"b":237},"2 points","1 segment",{"a":239,"b":240},"3 points","3 segments",{"a":242,"b":243},"4 points","6 segments",{"a":245,"b":246},"5 points","10 segments",{"a":248,"b":249},"6 points","15 segments",{"a":251,"b":252},"8 points","28 segments",{"a":254,"b":255},"10 points","45 segments","Connect each number of marked points on a line with the number of segments they make.","Connect the pairs. The rule is segments = n × (n − 1) ÷ 2.\n\n- 2 points → 1 segment\n- 3 points → 3 segments\n- 4 points → 6 segments\n- 5 points → 10 segments\n- 6 points → 15 segments\n- 8 points → 28 segments (8 × 7 ÷ 2)\n- 10 points → 45 segments (10 × 9 ÷ 2)\n\nNotice the answers grow faster and faster: doubling the points from 5 to 10 more than quadruples the segments, from 10 to 45.",{"id":259,"type":260,"title":261,"problem":262,"steps":263},"we-bus","worked_example","Tickets on a bus route","A straight bus route in a town has 7 stops. The conductor has a different ticket for every possible journey between two stops, and a ticket from stop 2 to stop 5 is the same as one from stop 5 to stop 2 (same fare). How many different tickets are needed? What if tickets are one-way only?",[264,265,266,267],"Each journey is a pair of stops, which is a segment on the line of the route.","Segments with 7 points: 7 × 6 ÷ 2 = **21** different tickets.","If direction matters, each segment gives two journeys: 21 × 2 = **42** tickets.","Real railway ticket tables work the same way: a fare chart for 7 stations has 21 different fares.",{"id":269,"type":164,"itemId":270,"prompt":271,"check":272,"hints":274,"feedback":276},"prac-seg-six","lines.investigate-seg-six","Six points are marked on a straight line. How many line segments have both end points among them?",{"kind":168,"answer":273,"tolerance":170},15,[275],"Use n × (n − 1) ÷ 2.",{"correct":277,"incorrect":278},"Right: 6 × 5 ÷ 2 = 15.","6 × 5 = 30 ends of segments; each segment has 2 ends, so 30 ÷ 2 = **15**.",{"id":280,"type":43,"markdown":281},"ray-count","What about **rays**? Count only rays that start at a marked point and pass through at least one other marked point. With points in a row, the two end points each start **1** such ray (pointing inward), and every point in the middle starts **2** (one each way). So with n points there are 1 + 1 + 2 × (n − 2) = **2 × (n − 1)** rays. With 5 points: 2 × 4 = 8 rays.",{"id":283,"type":164,"itemId":284,"prompt":285,"check":286,"hints":288,"feedback":291},"prac-rays-five","lines.investigate-rays-count","Points A, B, C, D lie on a line in that order. How many different rays start at one of these points and pass through another one of them?",{"kind":168,"answer":287,"tolerance":170},6,[289,290],"A and D each start one ray pointing inward.","B and C each start two rays, one in each direction.",{"correct":292,"incorrect":293},"Yes: 1 + 2 + 2 + 1 = 6.","A: ray AB. B: rays BA and BC. C: rays CB and CD. D: ray DC. Total **6** = 2 × (4 − 1). Note that ray AB, ray AC and ray AD are all the same ray.",{"id":295,"type":121,"title":296,"eyebrow":297,"navLabel":298},"ch03","Hidden segments in figures","Chapter 03","3 Hidden segments",{"id":300,"type":43,"markdown":301},"hidden-intro","Segments hide inside figures, and counting them carefully is a classic puzzle. The trick is to be systematic: count the segments **on each straight line** of the figure using the n × (n − 1) ÷ 2 rule, then add up the lines.",{"id":303,"type":260,"title":304,"problem":305,"steps":306},"we-triangle-cevian","A triangle with one extra line","Triangle ABC has a point D on side BC, and segment AD is drawn. How many segments can you find in the figure, using only the points A, B, C and D?",[307,308,309,310],"List the straight lines in the figure: line BC (with points B, D, C), line AB, line AC and line AD.","On line BC there are 3 marked points, giving 3 × 2 ÷ 2 = **3** segments: BD, DC and BC.","Lines AB, AC and AD each have only 2 marked points, so 1 segment each: **3** more.","Total: 3 + 3 = **6** segments. People who count only the \"pieces they can see\" usually find 5 and miss BC as a whole.",{"id":312,"type":127,"prompt":313,"options":314,"explanation":319},"predict-fan","From corner A of a triangle, three segments are drawn to points on the opposite side BC, which now carries B, three new points and C. How many segments are there on side BC alone?",[315,316,317,318],{"id":131,"label":97},{"id":133,"label":102},{"id":135,"label":103},{"id":137,"label":106},"**10.** Side BC now carries 5 marked points, and every pair of them makes a segment: 5 × 4 ÷ 2 = 10. Only 4 of them are the small pieces you see between neighbouring points; the other 6 are made of two or more pieces joined end to end.",{"id":321,"type":164,"itemId":322,"prompt":323,"check":324,"hints":325,"feedback":328},"prac-fan-total","lines.investigate-fan-total","In the figure from the prediction (corner A joined to B, C and three points on BC), how many segments are there altogether?",{"kind":168,"answer":273,"tolerance":170},[326,327],"Side BC gives 10. Now count segments from A.","A is joined to 5 points on BC.",{"correct":329,"incorrect":330},"Right: 10 on BC + 5 from A = 15.","On BC: 5 × 4 ÷ 2 = 10. From A: one segment to each of the 5 points on BC, that is 5. Total **15**.",{"id":332,"type":121,"title":333,"eyebrow":334,"navLabel":335},"ch04","How many crossings can lines make?","Chapter 04","4 Crossing points",{"id":337,"type":127,"prompt":338,"options":339,"explanation":347},"predict-three-lines","Draw three straight lines on a page. Which numbers of crossing points are possible? Pick the fullest correct list.",[340,342,343,345],{"id":131,"label":341},"Only 3",{"id":133,"label":157},{"id":135,"label":344},"0, 1, 2 or 3",{"id":137,"label":346},"0, 1, 2, 3 or 4","**0, 1, 2 or 3.** All three parallel: 0 crossings. All three through one point (concurrent): 1. Two parallel and the third cutting both: 2. Otherwise: 3, making a small triangle. Four is impossible, because each pair of lines can cross at most once and there are only 3 pairs.",{"id":349,"type":43,"markdown":350},"four-lines","With **four** lines there are 6 pairs, so at most 6 crossing points. But not every number from 0 to 6 is possible. A computer check of many random arrangements finds 0, 1, 3, 4, 5 and 6 crossing points, and never exactly 2. Try to see why 2 is impossible: if two lines cross, a third line that crosses either of them must add a new point unless it goes through the first crossing, and then the fourth line faces the same problem.",{"id":352,"type":82,"caption":353,"columns":354,"rows":359},"table-crossings","Crossing points of n lines in a plane",[355,356,357,358],"Lines","Pairs of lines","Most crossings","Numbers of crossings that are possible",[360,362,364,366],[94,91,91,361],"0, 1",[96,96,96,363],"0, 1, 2, 3",[97,99,99,365],"0, 1, 3, 4, 5, 6",[102,103,103,367],"0, 1, 4, 5, 6, 7, 8, 9, 10",{"id":369,"type":229,"component":370,"componentVersion":5,"config":371,"objective":421,"textAlternative":422},"lab-sort-possible","sort-game",{"prompt":372,"bins":373,"items":380,"seconds":170},"Possible or impossible? Think about pairs of lines.",[374,377],{"id":375,"label":376},"possible","Possible",{"id":378,"label":379},"impossible","Impossible",[381,385,389,393,397,401,405,409,413,417],{"id":382,"label":383,"bin":375,"why":384},"l3p0","3 lines with 0 crossing points","Make all three parallel.",{"id":386,"label":387,"bin":375,"why":388},"l3p2","3 lines with exactly 2 crossing points","Two parallel lines and a third cutting both.",{"id":390,"label":391,"bin":378,"why":392},"l3p4","3 lines with 4 crossing points","Only 3 pairs of lines, and each pair crosses at most once.",{"id":394,"label":395,"bin":375,"why":396},"l4p6","4 lines with 6 crossing points","No two parallel and no three through one point.",{"id":398,"label":399,"bin":378,"why":400},"l4p2","4 lines with exactly 2 crossing points","A computer check of every arrangement type never finds 2.",{"id":402,"label":403,"bin":375,"why":404},"l4p1","4 lines with exactly 1 crossing point","All four concurrent, like spokes through a hub.",{"id":406,"label":407,"bin":378,"why":408},"p4l5","4 points making exactly 5 lines through pairs","Four points make 1, 4 or 6 lines only.",{"id":410,"label":411,"bin":375,"why":412},"p4l4","4 points making exactly 4 lines","Put exactly three of them on one line.",{"id":414,"label":415,"bin":378,"why":416},"two-twice","Two straight lines crossing at 2 points","Two points fix one line, so two different lines share at most one point.",{"id":418,"label":419,"bin":378,"why":420},"p3l2","3 points making exactly 2 lines","Three points give 1 line (collinear) or 3 lines.","Decide which arrangements of points and lines are possible.","Ten claims about points and lines. Sort each as possible or impossible.\n\n**Possible:** 3 lines with 0 crossings (all parallel); 3 lines with exactly 2 crossings (two parallel, one cutting both); 4 lines with 6 crossings (general position); 4 lines with exactly 1 crossing (all concurrent); 4 points making exactly 4 lines (exactly three collinear).\n\n**Impossible:** 3 lines with 4 crossings (only 3 pairs); 4 lines with exactly 2 crossings (never happens); 4 points making exactly 5 lines (only 1, 4 or 6); two straight lines crossing twice; 3 points making exactly 2 lines.",{"id":424,"type":121,"title":425,"eyebrow":426,"navLabel":427},"ch05","Measuring experiments and tricks on the eye","Chapter 05","5 Measuring test",{"id":429,"type":43,"markdown":430},"measure-exp","Here is an experiment for two or more people. Draw one segment, about 7 to 8 cm long, on a sheet of paper. Without telling each other your answers, everyone measures it three ways:\n\n1. With a ruler, looking at it from **the side** (sitting low, eye near the desk).\n2. With a ruler, eye **directly above** each end.\n3. With a **divider**, then reading the divider on the ruler from directly above.\n\nWrite each result to the nearest millimetre. Then compare: which method gives readings closest to each other?",{"id":432,"type":82,"caption":433,"columns":434,"rows":439},"table-measure-sample","Example readings from a class trial (illustrative: your numbers will differ)",[435,436,437,438],"Person","Ruler, from the side","Ruler, from above","Divider",[440,444,447,451,454],[441,442,443,443],"Anu","7.5 cm","7.3 cm",[445,446,443,443],"Bilal","7.1 cm",[448,449,450,443],"Chitra","7.6 cm","7.4 cm",[452,453,443,443],"Dev","7.2 cm",[455,456,457,458],"Spread (largest − smallest)","0.5 cm","0.1 cm","0.0 cm",{"id":460,"type":76,"variant":116,"title":461,"markdown":462},"obs-spread","What the spread tells you","In these example readings, looking from the side scatters the answers by half a centimetre, some too big and some too small depending on which side each person sat. Looking from directly above shrinks the spread to 1 mm. The divider, whose sharp points touch the paper, removes parallax completely. A small spread means a method is **reliable**; it does not by itself prove the method is **accurate**, but agreement between careful methods is good evidence.",{"id":464,"type":127,"prompt":465,"options":466,"explanation":473},"predict-thick-ruler","A thick wooden ruler lies flat on a segment. You read it from the side, your eye to the **right** of the mark. Compared with the true value, will your reading of the right-hand end be…",[467,469,471],{"id":131,"label":468},"Too small",{"id":133,"label":470},"Too big",{"id":135,"label":472},"Exactly right","**Too big.** The scale is raised above the paper. Looking from the right, your line of sight to the end of the segment passes through the top of the ruler a little further to the right, so you read a slightly larger number. Looking from the left gives a reading that is too small. That is why the side readings in the table scatter both ways.",{"id":475,"type":76,"variant":476,"title":477,"markdown":478},"tryit-thread","try_it","Measuring a curve with thread","Rulers only measure straight things. To measure a curve, such as the outline of a bangle or a curved road on a map, lay a piece of thread carefully along it, mark where it ends, then straighten the thread along a ruler. You have turned the curve into a segment. Try measuring the edge of a ₹5 coin this way and compare with a friend.",{"id":480,"type":43,"markdown":481},"illusion-intro","Comparing segments **by observation** feels natural, but the brain uses shortcuts that can fool it. Test these on yourself before you measure.",{"id":483,"type":127,"prompt":484,"options":485,"explanation":492},"predict-t","Draw an upside-down T: a horizontal segment 6 cm long, and from its midpoint a vertical segment going up, also 6 cm long. Which looks longer?",[486,488,490],{"id":131,"label":487},"The vertical one",{"id":133,"label":489},"The horizontal one",{"id":135,"label":491},"They look equal","Most people see the **vertical** one as longer, although both are 6 cm. This is the **horizontal–vertical illusion**. The effect is strengthened because the vertical segment starts in the middle of the horizontal one and so seems to cut it into two shorter halves. Measure both to convince yourself.",{"id":494,"type":495,"title":496,"prompt":497,"options":498},"explorer-illusions","explorer","Three famous tricks on the eye","Pick an illusion to see what it does and how to beat it.",[499,511,520],{"id":500,"label":501,"chain":502,"badge":507,"note":510},"muller","Müller-Lyer",[503,504,505,506],"Two equal segments","Arrowheads out \u003C >","Arrowheads in > \u003C","Second looks longer",{"text":508,"tone":509},"Equal lengths","no","The fins at the ends make the brain misjudge where each segment stops. Cover the fins with your fingers, or use a divider, and the lengths match exactly.",{"id":512,"label":513,"chain":514,"badge":518,"note":519},"hv","Horizontal–vertical",[515,516,517],"Upside-down T","Both strokes equal","Upright looks longer",{"text":508,"tone":509},"Our eyes tend to overestimate vertical distances. Turn the page sideways and the illusion swaps round, which proves it is in your head, not on the paper.",{"id":521,"label":522,"chain":523,"badge":527,"note":528},"ponzo","Ponzo (railway)",[524,525,526],"Two rails meeting in the distance","Two equal bars across them","Top bar looks longer",{"text":508,"tone":509},"Draw two lines closing in like railway tracks in a photo, then two equal horizontal bars, one near the top and one near the bottom. The brain reads the top as further away, so it guesses the top bar must be bigger. Artists use the same effect for perspective.",{"id":530,"type":76,"variant":531,"title":532,"markdown":533},"mis-seeing","misconception","“Seeing is believing”","In geometry, seeing is a good way to get **ideas** and a bad way to **check** them. Always back up a comparison by tracing or measuring. The same goes for parallel lines: two lines can look parallel and meet a metre off the page.",{"id":535,"type":121,"title":536,"eyebrow":537,"navLabel":538},"ch06","Always, sometimes or never?","Chapter 06","6 Always or never",{"id":540,"type":43,"markdown":541},"asn-intro","A statement in geometry can be **always true**, **sometimes true** (true in some cases and false in others) or **never true**. To show something is *not always* true, one counterexample is enough. To show something is *always* true, you need a reason that covers every case.",{"id":543,"type":229,"component":370,"componentVersion":5,"config":544,"objective":605,"textAlternative":606},"lab-sort-asn",{"prompt":545,"bins":546,"items":556,"seconds":170},"Is each statement always, sometimes or never true?",[547,550,553],{"id":548,"label":549},"always","Always",{"id":551,"label":552},"sometimes","Sometimes",{"id":554,"label":555},"never","Never",[557,561,565,569,573,577,581,585,589,593,597,601],{"id":558,"label":559,"bin":548,"why":560},"two-points","Exactly one line passes through two different points.","Two points fix a line.",{"id":562,"label":563,"bin":551,"why":564},"three-collinear","Three points are collinear.","Beads on a thread, yes; corners of a triangle, no.",{"id":566,"label":567,"bin":554,"why":568},"meet-twice","Two different straight lines meet at two points.","They meet at most once.",{"id":570,"label":571,"bin":554,"why":572},"parallel-meet","Two parallel lines in a plane meet if extended far enough.","That is exactly what parallel rules out.",{"id":574,"label":575,"bin":551,"why":576},"perp-hv","Two perpendicular lines are one horizontal and one vertical.","A tilted square corner is perpendicular with neither line level.",{"id":578,"label":579,"bin":548,"why":580},"seg-length","A line segment has a length.","Two end points give a measurable length.",{"id":582,"label":583,"bin":554,"why":584},"ray-same","Ray AB is the same as ray BA.","They start at different points and point opposite ways.",{"id":586,"label":587,"bin":548,"why":588},"perp-four","When two lines are perpendicular, all four angles are right angles.","If one is 90°, the others are 180° − 90° = 90°.",{"id":590,"label":591,"bin":551,"why":592},"skew-plane","Two lines that never meet are parallel.","In a plane, yes; in space they may be skew.",{"id":594,"label":595,"bin":551,"why":596},"concurrent3","Three lines meet at a single point.","Concurrent lines do; three lines forming a triangle do not.",{"id":598,"label":599,"bin":548,"why":600},"midpoint-one","A segment has exactly one midpoint.","There is only one point halfway between the two ends.",{"id":602,"label":603,"bin":554,"why":604},"line-ends","A line has an end point.","A line goes on forever both ways.","Sort statements about lines into always, sometimes and never true.","Twelve statements to sort.\n\n**Always true:** exactly one line through two points; a segment has a length; perpendicular lines make four right angles; a segment has exactly one midpoint.\n\n**Sometimes true:** three points are collinear; perpendicular lines are horizontal and vertical; two lines that never meet are parallel (true in a plane, false in space, where they may be skew); three lines meet at a single point.\n\n**Never true:** two different lines meeting twice; parallel lines in a plane meeting; ray AB equal to ray BA; a line having an end point.\n\nFor every \"sometimes\", find one example where it is true and one where it is false.",{"id":608,"type":164,"itemId":609,"prompt":610,"check":611,"hints":623,"feedback":626},"prac-counterexample","lines.investigate-counterexample","Sanya says: \"If two lines never meet, they must be parallel.\" Which is a counterexample?",{"kind":612,"options":613,"correct":622},"choice",[614,616,618,620],{"id":131,"label":615},"Two railway rails",{"id":133,"label":617},"Two notebook lines",{"id":135,"label":619},"The edge where the front wall meets the ceiling, and the edge where a side wall meets the floor",{"id":137,"label":621},"Two lines of a plus sign",[135],[624,625],"A counterexample must never meet **and** not be parallel.","Think about edges in a room, not lines on a page.",{"correct":627,"incorrect":628},"Yes: those edges never meet and are not parallel. They are skew lines.","The room edges in (c) never meet but run in different directions at different heights. They are **skew**, so Sanya's rule fails in 3D. It only works for lines in one plane.",{"id":630,"type":121,"title":631,"eyebrow":632,"navLabel":633},"ch07","Investigating with paper folds","Chapter 07","7 Paper folding",{"id":635,"type":43,"markdown":636},"fold-intro","A sheet of paper is a surprisingly powerful geometry tool. Every fold is a perfectly straight line, and folding a crease onto itself makes an exact right angle. Try these folds on scrap paper; each one is an investigation.",{"id":638,"type":47,"title":639,"items":640},"steps-folds","Four folding investigations",[641,645,649,653],{"title":642,"tag":643,"text":644},"A perpendicular","fold a crease onto itself","Fold once to make crease l. Fold again so that l lies on itself. The new crease is perpendicular to l. Check with a set square.",{"title":646,"tag":647,"text":648},"A parallel","two perpendiculars","Make two different creases, each perpendicular to l. What do you notice about these two creases? Measure the gap between them at both ends.",{"title":650,"tag":651,"text":652},"Perpendicular bisector","end onto end","Draw segment AB. Fold so A lands on B. The crease crosses AB at its midpoint, at 90°.",{"title":654,"tag":655,"text":656},"Three bisectors","a triangle","Draw a triangle. Fold the perpendicular bisector of each side. Do the three creases meet at one point?",{"id":658,"type":127,"prompt":659,"options":660,"explanation":667},"predict-two-perps","You fold two creases, both perpendicular to the same crease l. What will the two new creases be?",[661,663,665],{"id":131,"label":662},"Parallel to each other",{"id":133,"label":664},"Perpendicular to each other",{"id":135,"label":666},"They will meet on l","**Parallel.** Both creases meet l at 90°, so they point in exactly the same direction. Lines in the same plane that point the same way and are not the same line never meet. Measure the gap at the top and bottom: it is equal. This is how carpenters draw parallel lines with a try square.",{"id":669,"type":76,"variant":670,"title":671,"markdown":672},"aha-concurrent","aha","The three creases always meet","When you fold the perpendicular bisectors of the three sides of any triangle, the three creases are **concurrent**: they pass through one point. It works for thin triangles, fat triangles and right-angled ones. Why? A point on the first crease is equally far from A and B; a point on the second is equally far from B and C. Where they cross, the point is equally far from all three corners, so it must also be on the third crease. That point is the centre of the circle through all three corners.",{"id":674,"type":675,"conceptId":676,"relation":677,"explanation":678},"conn-constructing-i","connection","constructing-angles","applied_in","Folding a perpendicular bisector is the paper version of the ruler-and-compass construction.",{"id":680,"type":121,"title":681,"eyebrow":682,"navLabel":683},"ch08","Grids of parallel lines","Chapter 08","8 Grids",{"id":685,"type":43,"markdown":686},"grid-intro","Draw 3 parallel lines across a page, then 4 parallel lines crossing them at right angles. You have made a small grid, like a window with glass panes or the lines of a badminton court. How many crossing points are there? How many rectangles can you find, counting big ones as well as small ones?",{"id":688,"type":127,"prompt":689,"options":690,"explanation":695},"predict-grid-points","3 horizontal parallel lines are crossed by 4 vertical parallel lines. How many crossing points?",[691,692,693,694],{"id":131,"label":110},{"id":133,"label":33},{"id":135,"label":99},{"id":137,"label":111},"**12.** Parallel lines never cross each other, so every crossing is between one horizontal and one vertical line: 3 × 4 = 12. Adding the lines (7) or counting pairs of all lines (21) are the tempting wrong answers.",{"id":697,"type":43,"markdown":698},"grid-rectangles","Now count rectangles. Every rectangle in the grid is fixed by choosing **2 of the horizontal lines** (its top and bottom) and **2 of the vertical lines** (its left and right sides). So:\n\n**rectangles = (pairs of horizontal lines) × (pairs of vertical lines)**.\n\nWith 3 horizontal and 4 vertical lines: 3 pairs × 6 pairs = **18** rectangles, although only 6 small panes are visible.",{"id":700,"type":82,"caption":701,"columns":702,"rows":708},"table-grid-rect","Rectangles in grids of parallel lines",[703,704,705,706,707],"Horizontal lines","Vertical lines","Crossing points","Small panes","All rectangles",[709,710,712,714,716,719],[94,94,97,91,91],[96,96,711,97,711],"9",[96,97,33,99,713],"18",[97,97,104,711,715],"36",[102,102,717,104,718],"25","100",[711,711,720,113,721],"81","1296",{"id":723,"type":260,"title":724,"problem":725,"steps":726},"we-chessboard","How many rectangles on a chessboard?","A chessboard has 8 × 8 small squares. How many rectangles of all sizes can you find on it (squares count as rectangles)?",[727,728,729,730],"The board is drawn with 9 horizontal and 9 vertical lines (8 squares need 9 lines).","Pairs of horizontal lines: 9 × 8 ÷ 2 = 36. The same for vertical lines: 36.","Each rectangle is a choice of one horizontal pair and one vertical pair: 36 × 36 = **1,296** rectangles.","Of these, the squares number 1² + 2² + … + 8² = 204. The rest, 1,092, are non-square rectangles.",{"id":732,"type":164,"itemId":733,"prompt":734,"check":735,"hints":737,"feedback":740},"prac-grid-rect","lines.investigate-grid-rect","A window has 4 horizontal bars and 5 vertical bars, forming a grid (count the frame edges as bars). How many rectangles of all sizes does the grid contain?",{"kind":168,"answer":736,"tolerance":170},60,[738,739],"Choose 2 horizontal bars and 2 vertical bars.","Pairs from 4 is 6; pairs from 5 is 10.",{"correct":741,"incorrect":742},"Yes: 6 × 10 = 60.","Pairs of horizontal bars: 4 × 3 ÷ 2 = 6. Pairs of vertical bars: 5 × 4 ÷ 2 = 10. Rectangles = 6 × 10 = **60**.",{"id":744,"type":121,"title":745,"eyebrow":746,"navLabel":747},"ch09","Clock hands: perpendicular and in a line","Chapter 09","9 Clock hands",{"id":749,"type":43,"markdown":750},"clock-intro","The two hands of a clock are like two rays from the same end point, the centre. As they turn, they are sometimes **perpendicular** (at 90° to each other), sometimes **opposite** (forming one straight line through the centre) and sometimes **on top of each other**. How often does each happen?",{"id":752,"type":127,"prompt":753,"options":754,"explanation":762},"predict-clock","Between 12 noon and 12 midnight, how many times are the hour hand and the minute hand exactly perpendicular?",[755,757,758,760],{"id":131,"label":756},"2 (at 3:00 and 9:00)",{"id":133,"label":33},{"id":135,"label":759},"22",{"id":137,"label":761},"24","**22.** Many people answer 2, because 3:00 and 9:00 are the only times it happens exactly on the hour. But the minute hand gains 5.5° on the hour hand every minute, so it passes through a 90° gap twice in nearly every hour: once while catching up and once after overtaking. Over 12 hours the minute hand gains 11 full turns, and each turn gives 2 right-angle moments: 11 × 2 = 22.",{"id":764,"type":82,"caption":765,"columns":766,"rows":770},"table-clock","When are the hands perpendicular? (computed exactly; times rounded to 0.01 minute)",[767,768,769],"Moment","Time after 12:00","Angle between hands",[771,775,779,782,785,788],[772,773,774],"1st","12:16.36","90°",[776,777,778],"2nd","12:49.09","90° (270° the other way)",[780,781,774],"3rd","1:21.82",[783,784,774],"4th","1:54.55",[786,787,774],"5th","2:27.27",[789,790,791],"6th","3:00.00","90° exactly on the hour",{"id":793,"type":794,"tone":795,"items":796},"spec-clock","spec","neutral",[797,801,805,809,813],{"label":798,"big":799,"value":800},"Minute hand","6° per min","360° in 60 minutes.",{"label":802,"big":803,"value":804},"Hour hand","0.5° per min","30° per hour, 360° in 12 hours.",{"label":806,"big":807,"value":808},"Gain","5.5° per min","The minute hand gains 6 − 0.5 = 5.5° every minute.",{"label":810,"big":811,"value":812},"Perpendicular","22 in 12 h","44 times a day.",{"label":814,"big":815,"value":816},"Opposite (a line)","11 in 12 h","Only 6:00 is exactly on the hour.",{"id":818,"type":164,"itemId":819,"prompt":820,"check":821,"hints":823,"feedback":826},"prac-clock-line","lines.investigate-clock-line","How many times in 12 hours do the two hands of a clock form one straight line **pointing in opposite directions** (like at 6:00)?",{"kind":168,"answer":822,"tolerance":170},11,[824,825],"The minute hand gains one full turn on the hour hand 11 times in 12 hours.","Each gain of a full turn passes through \"opposite\" once.",{"correct":827,"incorrect":828},"Yes: 11 times in 12 hours, 22 times a day.","The minute hand gains 360° on the hour hand 11 times in 12 hours, passing through the opposite position once each time: **11** times.",{"id":830,"type":121,"title":831,"eyebrow":832,"navLabel":833},"ch10","Field work: lines in the real world","Chapter 10","10 Field work",{"id":835,"type":43,"markdown":836},"survey-intro","Take a notebook round your school or home and record every example you can find of each relationship. Then look at the evidence: which kind is most common? Why do builders and designers use so many right angles and parallel lines?",{"id":838,"type":82,"caption":839,"columns":840,"rows":845},"table-survey","A lines survey sheet (fill in your own)",[841,842,843,844],"Place","Parallel lines","Perpendicular lines","Intersecting (not 90°)",[846,851,856,861,866],[847,848,849,850],"Classroom","Edges of the blackboard, top and bottom","Door frame and floor","The legs of an easel",[852,853,854,855],"Playground","Lanes on the running track (straight part)","Goal post and crossbar","Slide and ladder of a swing set",[857,858,859,860],"Kitchen","Shelves","Tiles on the wall","The blades of a pair of tongs",[862,863,864,865],"Road","Lane markings","The crossing path and the road edge","A Y-shaped junction",[867,868,868,868],"Your home","…",{"id":870,"type":76,"variant":116,"title":871,"markdown":872},"obs-rightangles","Why so many right angles and parallels?","Right angles and parallel lines make shapes that **fit together** without gaps, **stack** neatly and are **easy to measure**. A rectangular brick sits flat on the one below it; shelves parallel to the floor keep things from sliding; walls perpendicular to the ground carry weight straight down. Slanted lines appear where they do a job: roof slopes shed monsoon rain, and diagonal braces stop frames from wobbling.",{"id":874,"type":229,"component":875,"componentVersion":5,"config":876,"objective":880,"textAlternative":881},"lab-spot-pairs-i","line-spotter",{"modes":877,"rounds":879},[878],"pairs",12,"Classify twelve pairs of lines, turned at random angles, quickly and accurately.","Twelve rounds of pairs. For each pair decide: **parallel**, **perpendicular** or **intersecting**.\n\nInvestigator's tips:\n\n- For **parallel**, imagine measuring the gap at both ends. Parallel lines keep exactly the same gap; if two lines cross anywhere in the picture they cannot be parallel.\n- For **perpendicular**, imagine fitting a folded-paper corner into the crossing. If it fits exactly, it is 90°. Intersecting pairs in this game cross at clearly slanted angles, between about 25° and 65° (or the matching wide angles).\n- Lines that are rotated on the screen can still be parallel or perpendicular. Direction on the page does not matter; the relationship between the two lines does.\n\nTry to beat your score from earlier layers while keeping your streak going.",{"id":883,"type":121,"title":884,"eyebrow":885,"navLabel":886},"ch11","Counting on courts, tracks and crossings","Chapter 11","11 Courts and tracks",{"id":888,"type":43,"markdown":889},"courts-count-intro","Real line markings make great counting problems. To keep the counting clean, we use **simplified models**: every lengthwise line runs the full length and every crosswise line runs the full width, so each lengthwise line crosses each crosswise line exactly once. Real courts are close to this but not exact, so treat the numbers as answers about the model.",{"id":891,"type":260,"title":892,"problem":893,"steps":894},"we-track-sleepers","Rails and sleepers","A straight stretch of track has 2 parallel rails. Across them lie 40 sleepers, each perpendicular to the rails. Treating rails and sleepers as lines, how many crossing points are there? How many rectangles of any size are there between the rails?",[895,896,897,898],"Rails never cross each other; sleepers never cross each other (they are parallel too). Every crossing is a rail with a sleeper: 2 × 40 = **80** crossing points.","A rectangle needs 2 rails (only 1 way to choose them) and 2 sleepers.","Pairs of sleepers: 40 × 39 ÷ 2 = 780.","Rectangles = 1 × 780 = **780**, although only 39 small ones are visible.",{"id":900,"type":260,"title":901,"problem":902,"steps":903},"we-kabaddi-model","A simplified kabaddi court","Model a kabaddi court with 4 lengthwise lines (2 side lines and 2 lobby lines) and 7 crosswise lines (2 end lines, 2 bonus lines, 2 baulk lines and the mid line), all running the full length or width. How many crossing points and how many rectangles does the model have?",[904,905,906,907],"Crossing points: each of the 4 lengthwise lines crosses each of the 7 crosswise lines once: 4 × 7 = **28**.","Rectangles: pairs of lengthwise lines 4 × 3 ÷ 2 = 6; pairs of crosswise lines 7 × 6 ÷ 2 = 21.","Rectangles = 6 × 21 = **126**.","On a real court the lobby lines and some crosswise lines do not run everywhere, so the real count is a little different: always say what your model assumes.",{"id":909,"type":164,"itemId":910,"prompt":911,"check":912,"hints":914,"feedback":916},"prac-zebra-count","lines.investigate-zebra-count","A zebra crossing has 8 white stripes. Each stripe is a rectangle with 4 sides. Treating every side as a segment, how many of these segments run **along** the road (parallel to the traffic)?",{"kind":168,"answer":913,"tolerance":170},16,[915],"Each stripe has 2 long sides along the road and 2 short sides across it.",{"correct":917,"incorrect":918},"Yes: 8 × 2 = 16 long sides run along the road.","Each stripe has 2 long sides running along the road (parallel to the traffic): 8 × 2 = **16**. There are 32 sides in total, and the other 16 run across the road.",{"id":920,"type":127,"prompt":921,"options":922,"explanation":927},"predict-extra-point","Five points are marked on a line and every segment between them is drawn. One more point is added to the line. How many **new** segments appear?",[923,924,925,926],{"id":131,"label":91},{"id":133,"label":102},{"id":135,"label":99},{"id":137,"label":106},"**5.** The new point makes one new segment with each of the 5 old points. The total goes from 10 to 15, and 15 − 10 = 5. This is exactly the \\\"increase\\\" column in the counting table earlier.",{"id":929,"type":76,"variant":531,"title":930,"markdown":931},"mis-count-visible","\\u201cCount only the pieces you can see\\u201d","On a line with points A, B, C, D, students often count 3 segments (AB, BC, CD), the visible pieces. They miss AC, BD and AD, which are made of pieces joined end to end. Fix: for each point, count the segments that **start** there and go to the right. A: 3, B: 2, C: 1. Total 6.",{"id":933,"type":260,"title":934,"problem":935,"steps":936},"we-left-to-right","The start-here method","Seven stations lie on a straight railway line. Count the segments between stations using the start-here method.",[937,938,939,940],"From the 1st station, segments go to each of the 6 stations to its right: 6.","From the 2nd: 5. From the 3rd: 4. Then 3, 2, 1, and 0 from the last station.","Total = 6 + 5 + 4 + 3 + 2 + 1 = **21**, the same as 7 × 6 ÷ 2.","This method never double-counts, because every segment is counted only from its left end.",{"id":942,"type":164,"itemId":943,"prompt":944,"check":945,"hints":947,"feedback":949},"prac-wires-poles","lines.investigate-wires-poles","Along a straight road, 12 electric poles stand in a line. A survey team wants to measure the distance between **every pair** of poles. How many distances must they measure?",{"kind":168,"answer":946,"tolerance":170},66,[948,275],"Each distance is a segment between two poles.",{"correct":950,"incorrect":951},"Right: 12 × 11 ÷ 2 = 66.","Every pair of poles is a segment: 12 × 11 ÷ 2 = **66** distances. (Only 11 are between neighbouring poles; the rest can be found by adding those.)",{"id":953,"type":76,"variant":116,"title":954,"markdown":955},"obs-add-neighbours","You only need to measure the neighbours","In the pole problem you do not really need all 66 measurements. Because the poles are collinear, the distance between any two is the sum of the neighbouring gaps between them (the betweenness rule AB + BC = AC). Eleven measurements are enough, and adding them gives the other 55. Measuring a few extra and checking the sums is how surveyors catch mistakes.",{"id":957,"type":229,"component":370,"componentVersion":5,"config":958,"objective":1009,"textAlternative":1010},"lab-sort-rule",{"prompt":959,"bins":960,"items":969,"seconds":170},"Which counting rule fits each situation?",[961,963,966],{"id":878,"label":962},"Pairs: n × (n − 1) ÷ 2",{"id":964,"label":965},"product","Product: a × b",{"id":967,"label":968},"rays","Rays: 2 × (n − 1)",[970,974,978,982,986,989,993,997,1001,1005],{"id":971,"label":972,"bin":878,"why":973},"stations","Tickets between 10 stations on one line (either direction same fare)","Each ticket is a pair of stations: 10 × 9 ÷ 2 = 45.",{"id":975,"label":976,"bin":964,"why":977},"rails-sleepers","Crossings of 2 rails with 30 sleepers","Every crossing is one rail with one sleeper: 2 × 30 = 60.",{"id":979,"label":980,"bin":878,"why":981},"handshakes","Handshakes among 12 players, each pair once","Each handshake is a pair: 12 × 11 ÷ 2 = 66.",{"id":983,"label":984,"bin":964,"why":985},"grid-points","Crossing points of 5 horizontal and 6 vertical grid lines","Parallel lines never cross, so count horizontal × vertical: 30.",{"id":280,"label":987,"bin":967,"why":988},"Rays that start at one of 6 points on a line and pass through another","End points start 1 ray each, the 4 middle points 2 each: 2 × 5 = 10.",{"id":990,"label":991,"bin":878,"why":992},"lines-general","Lines through 7 points with no three collinear","Each pair of points gives its own line: 21.",{"id":994,"label":995,"bin":878,"why":996},"crossings-general","Most crossing points of 8 lines","One crossing per pair of lines at most: 28.",{"id":998,"label":999,"bin":964,"why":1000},"zebra-sides","Long sides of 9 zebra stripes, 2 per stripe","9 stripes × 2 long sides = 18.",{"id":1002,"label":1003,"bin":878,"why":1004},"seg-line","Segments on a line with 9 marked points","Each segment is a pair of points: 36.",{"id":1006,"label":1007,"bin":967,"why":1008},"ray-count-4","Rays from 4 points on a line, each through another marked point","2 × 3 = 6.","Decide which counting rule fits each everyday counting situation.","Ten situations to sort by counting rule.\n\n**Pairs, n × (n − 1) ÷ 2:** tickets between 10 stations (45); handshakes among 12 players (66); lines through 7 points, no three collinear (21); most crossing points of 8 lines (28); segments on a line with 9 points (36).\n\n**Product, a × b:** crossings of 2 rails with 30 sleepers (60); crossing points of 5 horizontal and 6 vertical lines (30); long sides of 9 zebra stripes (18). Use the product when the things come from **two different families** that each meet the other once.\n\n**Rays, 2 × (n − 1):** rays from 6 points on a line (10); rays from 4 points (6).",{"id":1012,"type":127,"prompt":1013,"options":1014,"explanation":1023},"predict-divider-short","A segment is only 0.8 cm long. Measuring with a ruler read from the side can be off by about 1 mm. Roughly what fraction of the length could that error be?",[1015,1017,1019,1021],{"id":131,"label":1016},"About 1 part in 100",{"id":133,"label":1018},"About 1 part in 8",{"id":135,"label":1020},"About half",{"id":137,"label":1022},"No error at all","**About 1 part in 8.** 1 mm out of 8 mm is 1 ÷ 8, more than 12%. The same 1 mm on a 20 cm segment is only 1 part in 200. Parallax matters most for **short** segments, which is exactly where a divider helps most.",{"id":1025,"type":121,"title":1026,"eyebrow":1027,"navLabel":1028},"ch12","What we found","Chapter 12","12 Wrap-up",{"id":1030,"type":82,"caption":1031,"columns":1032,"rows":1036},"table-findings","Results of our investigations",[1033,1034,1035],"Question","Finding","Status",[1037,1041,1045,1049,1052,1056,1060],[1038,1039,1040],"Lines through n points, no three collinear","n × (n − 1) ÷ 2","Pattern, explained by counting pairs",[1042,1043,1044],"Segments on a line with n points","n × (n − 1) ÷ 2 (triangular numbers)","Pattern, explained by counting ends",[1046,1047,1048],"Rays from n points on a line","2 × (n − 1)","Explained: 1 from each end point, 2 from each inner point",[1050,344,1051],"Crossings of 3 lines","Checked every arrangement",[1053,1054,1055],"Crossings of 4 lines","0, 1, 3, 4, 5 or 6 (never 2)","Computer check; proof in Deepen",[1057,1058,1059],"Best measuring method","Divider or eye directly above","Experiment (compare spreads)",[1061,1062,1063],"Perpendicular bisectors of a triangle","Always concurrent","Folding evidence plus a reason",{"id":1065,"type":1066,"title":1067,"terms":1068},"glossary-investigate","glossary","Words for investigating",[1069,1073,1077,1081,1085,1089,1093],{"term":1070,"meaning":1071,"example":1072},"Conjecture","A guess based on a pattern, not yet proved.","\"Segments = n(n − 1) ÷ 2\" before we explain it",{"term":1074,"meaning":1075,"example":1076},"Counterexample","One example that shows a statement is not always true.","Skew lines break \"never meet ⇒ parallel\"",{"term":1078,"meaning":1079,"example":1080},"Triangular numbers","1, 3, 6, 10, 15…: the numbers of dots in growing triangles, and of segments on a line.","10 = 1 + 2 + 3 + 4",{"term":1082,"meaning":1083,"example":1084},"General position","Lines with no two parallel and no three through one point.","4 such lines make 6 crossings",{"term":1086,"meaning":1087,"example":1088},"Reliable","Giving nearly the same answer each time it is repeated.","Divider readings in the trial",{"term":1090,"meaning":1091,"example":1092},"Optical illusion","A picture that tricks the eye into a wrong judgement.","Müller-Lyer arrows",{"term":1094,"meaning":1095,"example":1096},"Circumcentre","The point where the perpendicular bisectors of a triangle's sides meet; equally far from all three corners.","Where the three folds meet",{"id":1098,"type":1099,"title":1100,"questions":1101},"quiz-investigate","quiz","Test your findings",[1102,1112,1121,1130,1140,1149,1161,1170,1181,1188],{"itemId":1103,"prompt":1104,"options":1105,"correct":135,"why":1111},"lines.investigate-q-clock","How many times in 12 hours are clock hands perpendicular?",[1106,1107,1109,1110],{"id":131,"label":94},{"id":133,"label":1108},"11",{"id":135,"label":759},{"id":137,"label":761},"The minute hand laps the hour hand 11 times, passing 90° twice per lap.",{"itemId":1113,"prompt":1114,"options":1115,"correct":133,"why":1120},"lines.investigate-q-four-lines","Four points, no three collinear. How many lines through pairs?",[1116,1117,1118,1119],{"id":131,"label":97},{"id":133,"label":99},{"id":135,"label":100},{"id":137,"label":33},"4 × 3 ÷ 2 = 6 pairs, each giving its own line.",{"itemId":1122,"prompt":1123,"options":1124,"correct":133,"why":1129},"lines.investigate-q-three-in-four","Four points with exactly three collinear make how many lines?",[1125,1126,1127,1128],{"id":131,"label":96},{"id":133,"label":97},{"id":135,"label":102},{"id":137,"label":99},"1 line for the collinear three, plus 3 lines from the fourth point.",{"itemId":1131,"prompt":1132,"options":1133,"correct":133,"why":1139},"lines.investigate-q-seg-eight","How many segments on a line with 8 marked points?",[1134,1135,1136,1137],{"id":131,"label":104},{"id":133,"label":210},{"id":135,"label":715},{"id":137,"label":1138},"56","8 × 7 ÷ 2 = 28.",{"itemId":1141,"prompt":1142,"options":1143,"correct":137,"why":1148},"lines.investigate-q-three-cross","Which number of crossings is impossible for three lines?",[1144,1145,1146,1147],{"id":131,"label":92},{"id":133,"label":91},{"id":135,"label":94},{"id":137,"label":97},"Three lines make only 3 pairs, each crossing at most once.",{"itemId":1150,"prompt":1151,"options":1152,"correct":133,"why":1160},"lines.investigate-q-reliable","Which method gave the most reliable readings in the trial?",[1153,1155,1156,1158],{"id":131,"label":1154},"Ruler read from the side",{"id":133,"label":438},{"id":135,"label":1157},"Guessing by eye",{"id":137,"label":1159},"Tracing","The divider's points touch the paper, so there is no parallax.",{"itemId":1162,"prompt":1163,"options":1164,"correct":133,"why":1169},"lines.investigate-q-rays","Five points on a line. How many rays start at one and pass through another?",[1165,1166,1167,1168],{"id":131,"label":102},{"id":133,"label":100},{"id":135,"label":103},{"id":137,"label":221},"2 × (5 − 1) = 8.",{"itemId":1171,"prompt":1172,"options":1173,"correct":133,"why":1180},"lines.investigate-q-two-perps","Two lines are each perpendicular to the same line l (all in one plane). They are…",[1174,1175,1176,1178],{"id":131,"label":664},{"id":133,"label":662},{"id":135,"label":1177},"Concurrent with l",{"id":137,"label":1179},"Skew","Both point in the direction at 90° to l, so they never meet.",{"itemId":1182,"prompt":1183,"options":1184,"correct":131,"why":1187},"lines.investigate-q-illusion","In the upside-down T illusion, which segment usually looks longer?",[1185,1186],{"id":131,"label":487},{"id":133,"label":489},"People tend to overestimate vertical lengths.",{"itemId":1189,"prompt":1190,"options":1191,"correct":131,"why":1200},"lines.investigate-q-counter","How many counterexamples do you need to show a statement is not always true?",[1192,1194,1196,1198],{"id":131,"label":1193},"One",{"id":133,"label":1195},"Two",{"id":135,"label":1197},"Ten",{"id":137,"label":1199},"You can never show it","A single case where it fails is enough.",{"id":1202,"type":1203,"prompt":1204},"reflect-investigate","reflection","Which result in this layer surprised you most? Write down the prediction you made, what actually happened, and how you would explain it to a friend.",{"id":1206,"type":1207,"title":1208,"points":1209},"cheat-investigate","summary","Cheat sheet",[1210,1211,1212,1213,1214,1215,1216,1217,1218],"Investigate by **asking, predicting, trying small cases, spotting a pattern, testing, explaining**.","A pattern is a **conjecture** until it is explained; one **counterexample** breaks it.","Lines through pairs of n points (no three collinear): **n × (n − 1) ÷ 2**. Four points give 1, 4 or 6 lines only.","Segments on a line with n points: **n × (n − 1) ÷ 2** = 1, 3, 6, 10, 15, 21, 28…","Rays from n points on a line (through another marked point): **2 × (n − 1)**.","Three lines cross in 0, 1, 2 or 3 points; four lines in 0, 1, 3, 4, 5 or 6 (never 2).","Measure with a divider or with your eye directly above the scale; side views scatter readings.","Eyes are fooled by fins, verticals and converging lines: **measure before you compare**.","Two lines perpendicular to the same line (in a plane) are **parallel**. The three perpendicular bisectors of a triangle are **concurrent**.",{"id":1220,"type":675,"conceptId":1221,"relation":1222,"explanation":1223},"conn-patterns-i","patterns","related_to","Counting segments gives the triangular numbers 1, 3, 6, 10, 15, a classic growing pattern.",{"id":1225,"type":675,"conceptId":1226,"relation":1227,"explanation":1228},"conn-angles-i","angles","helps_understand","Folding a crease onto itself makes a right angle, the reference for classifying every other angle.",{"id":1230,"type":1231,"sourceIds":1232},"sources-investigate","sources",[1233,1234,1235,1236,1237,1238,1239,1240],"lines-ncert-math-6","lines-ncert-ganita-prakash-6","lines-ncert-math-7","lines-mathsisfun-line","lines-mathsisfun-parallel-perpendicular","lines-wiki-skew-lines","lines-wiki-kabaddi","lines-irc-35-road-markings",[1233,1234,1235,1236,1237,1238,1239,1240],"needs_review",{"generatedBy":1244,"notes":1245},"claude-code","Draft generated by a scripted generator; every count and length was computed in Python. Pending owner review.","10cd59e6c868f592322638865d4ee9e3729476ac54b4e3099b6df1b3891b8a80",{"logic:practice":1248,"component:match-pairs@1":1249,"component:sort-game@1":1250,"component:line-spotter@1":1251,"source:lines-irc-35-road-markings":1252,"source:lines-mathsisfun-line":1253,"source:lines-mathsisfun-parallel-perpendicular":1254,"source:lines-ncert-ganita-prakash-6":1255,"source:lines-ncert-math-6":1256,"source:lines-ncert-math-7":1257,"source:lines-wiki-kabaddi":1258,"source:lines-wiki-skew-lines":1259},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","1279a4e23634aba0fd1cb2bb76638a44d6b592131c326e194fdd1bf593a12f93","f7987084cd045541cbd20cd4782d1f80d4737eabbdde880e6e812b7dacdff140","982f31ecb1e1e17e74ba13872e9e63e218010de6c5653901ebd0db8a12413bc3","ee8cac463d4577f5a05460d731e8e92e3d4268172d2221efc74a4e61e4376345","b7ec4502e5c7912ecccfeb371ab30a19043c5598016c6d51acbc678290fc40a2","219993d3ebc010eeac3a16481eda537da1557ef707d475cbca1ba05a49dfff80","f380f754917bdc6d17093f871ff43201664a75572d28b860df518fdbf8f740be","c3c59ba667c814a640c01ea119d156347dc04c3844e0067f4e60f9ffae877e8f","530f5f83c26d9065c72fab1d85bb3ea6ed8301d7b9faa7374e7e2296bc6ae626",{"state":1261,"reviewer":1262,"selfReview":1263,"reviewedAt":1264,"method":1265},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598475]