[{"data":1,"prerenderedAt":1244},["ShallowReactive",2],{"layer:lines:discover":3},{"layer":4,"contentHash":1222,"dependencyHashes":1223,"approval":1237,"releaseId":1243},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":17,"plate":18,"blocks":39,"sourceIds":1217,"reviewStatus":1218,"authoring":1219},1,"lines","en","discover","Straight paths: points, lines, rays and segments","Meet the alphabet of geometry in torch beams, railway tracks and cricket creases","Meet points, line segments, rays and lines through everyday things, then see how two lines can cross, meet at square corners or run side by side forever.",[13,14,15,16],"Tell a point, a line segment, a ray and a line apart by looking at their ends.","Name points with capital letters and name segments, rays and lines using two points.","Recognise intersecting, parallel and perpendicular lines in everyday objects.","Explain the difference between a line and a curve, describe a plane, and tell horizontal from vertical.",25,{"title":19,"rows":20},"Lesson plate",[21,24,27,30,33,36],{"label":22,"value":23},"Depth","Discover",{"label":25,"value":26},"Reading time","≈ 25 minutes",{"label":28,"value":29},"Prior knowledge","None",{"label":31,"value":32},"Chapters","12",{"label":34,"value":35},"Labs","Line spotter ×2, sort games ×2, memory match",{"label":37,"value":38},"Tools","Pencil, ruler, a scrap of paper",[40,44,50,55,61,64,85,90,93,98,103,106,126,130,133,143,163,168,183,194,202,207,212,215,219,239,257,262,267,270,291,296,299,303,333,389,393,407,427,432,435,448,479,482,486,491,496,499,503,523,534,539,542,546,569,573,578,581,585,589,613,625,636,652,657,660,679,683,701,706,709,712,749,761,820,883,888,891,914,922,936,938,942,945,949,968,985,989,994,1019,1023,1066,1171,1175,1189,1195,1199,1204],{"id":41,"type":42,"markdown":43},"intro-straight","prose","Look up from wherever you are sitting. How many **straight edges** can you count in ten seconds? The edge of a door. The top of a table. The lines in your notebook. The cable running along the wall. The edge of your phone.\n\nNow go outside in your mind. A railway track running off into the distance. The white crease line on a cricket pitch. A kite string pulled tight on Makar Sankranti. The beam of a torch at night. The border of a badminton court painted on the floor of a school hall.\n\nAll of these are made of the same few ideas: **points**, **lines**, **rays** and **line segments**. They are the alphabet of geometry. Every shape you will ever draw, from a triangle to a rangoli to the plan of a house, is built from them.",{"id":45,"type":46,"variant":47,"title":48,"markdown":49},"intro-how","callout","observation","How to use this lesson","This lesson is for meeting the ideas, not memorising rules. Each chapter starts with something you have seen a hundred times. When you meet a **prediction**, choose an answer before reading on. When you meet a **lab**, play a few rounds: the games are where the ideas get into your fingers.",{"id":51,"type":46,"variant":52,"title":53,"markdown":54},"intro-question","question","The big question","What is the difference between a line, a ray and a line segment? And why do railway tracks look as if they meet far away, even though the train would fall off if they really did?",{"id":56,"type":57,"title":58,"eyebrow":59,"navLabel":60},"ch01","chapter","A point: a place with no size","Chapter 01","1 A point",{"id":62,"type":42,"markdown":63},"point-dot","Put the sharp tip of your pencil on a page and press very lightly. You have made a tiny dot. That dot shows a **place** on the page.\n\nA **point** is exactly that: an exact position. The dot you drew is only a *picture* of a point. A real mathematical point is so small that it has **no length, no width and no thickness at all**. You cannot measure it. You can only say *where* it is.\n\nThat sounds strange, but you already use points like this. When a friend says \"meet me at the school gate\", you do not worry about how big the meeting place is. You only care *where* it is.",{"id":65,"type":66,"tone":67,"items":68},"spec-points","spec","blue",[69,73,77,81],{"label":70,"big":71,"value":72},"On a map","📍","A pin marks the exact place of your school. The pin has size; the place it points to does not.",{"label":74,"big":75,"value":76},"In the sky","★","A star looks like a point of light, even though it is a huge ball of gas, because it is so far away.",{"label":78,"big":79,"value":80},"On a ruler","0 cm","Each mark shows an exact point: 0, 1, 2 and so on. The mark has thickness only so that you can see it.",{"label":82,"big":83,"value":84},"In cricket","the stumps","The umpire imagines the exact point where the ball pitched when deciding an LBW.",{"id":86,"type":46,"variant":87,"title":88,"markdown":89},"def-point","definition","Point","A **point** marks an exact position. It has no size. We draw it as a small dot and name it with a **capital letter**, such as point A, point P or point Z.",{"id":91,"type":42,"markdown":92},"point-naming","Why capital letters? So that we can talk about points without pointing. If a page has three dots, saying \"that one, no, the other one\" is confusing. Saying \"point B\" is clear.\n\nMathematicians agreed on a simple habit: **points get capital letters**. You will see this in every textbook in India and around the world. You can use any capital letter you like, but each point on a page should have its own letter.",{"id":94,"type":46,"variant":95,"title":96,"markdown":97},"tryit-dots","try_it","Make a dot map of your room","Draw a rough plan of your room. Put a dot for the door, one for your bed's corner and one for the window. Name them D, B and W. Now write directions for a friend using only your letters: \"Walk from D towards W.\" Notice how much easier it is when each place has a name.",{"id":99,"type":57,"title":100,"eyebrow":101,"navLabel":102},"ch02","Joining two points: the line segment","Chapter 02","2 Line segments",{"id":104,"type":42,"markdown":105},"segment-intro","Mark two points on a page, A and B. Put your ruler against both of them and draw the straight path from A to B. What you have drawn is a **line segment**.\n\nA line segment is the **straight path between two points**. It starts at one point and stops at the other. Those two points are called its **end points**. Because it stops at both ends, a line segment has a length you can measure: 5 cm, 12 cm, 3.5 cm.",{"id":107,"type":66,"tone":108,"items":109},"spec-segments","amber",[110,114,118,122],{"label":111,"big":112,"value":113},"A pencil","≈ 17 cm","A new pencil is a good picture of a line segment: straight, with two ends.",{"label":115,"big":116,"value":117},"Cricket pitch","22 yards","From the stumps at one end to the stumps at the other is about 20.12 m.",{"label":119,"big":120,"value":121},"Edge of a notebook","two ends","Each edge of a page starts at one corner and stops at another.",{"label":123,"big":124,"value":125},"A stretched string","tight","A kite string pulled tight between your hand and the kite is almost a segment.",{"id":127,"type":46,"variant":87,"title":128,"markdown":129},"def-segment","Line segment","A **line segment** is the straight path between two points. It has **two end points** and a **fixed length**. Segment AB joins point A to point B. Segment AB and segment BA are the same segment.",{"id":131,"type":42,"markdown":132},"segment-shortest","Here is something surprising. Of all the paths you could take from A to B (wiggly ones, zigzag ones, curved ones), the **straight** one is the **shortest**. That is why people cut across a grassy corner of a park instead of walking along the path: their feet know that a straight line is the shortest way.",{"id":134,"type":135,"title":136,"problem":137,"steps":138},"we-shortest","worked_example","Which path is shortest?","Riya walks from her house (H) to the kirana shop (K). She can walk along two roads that meet at a corner: 300 m then 400 m. Or she can cut straight across an open ground from H to K, which is 500 m. How much shorter is the straight path?",[139,140,141,142],"Along the roads: 300 m + 400 m = 700 m.","Straight across: 500 m.","Difference: 700 − 500 = **200 m** shorter.","The straight path HK is a line segment. It is always the shortest path between two points on flat ground.",{"id":144,"type":145,"title":146,"items":147},"steps-draw-seg","steps","Drawing a segment 6 cm long",[148,151,154,157,160],{"title":149,"text":150},"Mark the start","Put a dot on the page and call it A.",{"title":152,"text":153},"Line up the zero","Place the ruler so that the 0 mark sits exactly on A. Many rulers have a little gap before the 0, so do not use the very end of the ruler.",{"title":155,"text":156},"Find 6","Run your eye along the ruler to the 6 cm mark and put a second dot there. Call it B.",{"title":158,"text":159},"Join them","Hold the ruler firmly and draw from A to B along its edge.",{"title":161,"text":162},"Label","Write \"AB = 6 cm\" beside your segment.",{"id":164,"type":46,"variant":165,"title":166,"markdown":167},"careful-zero","careful","Start at the zero, not the end","On most rulers the 0 mark is a few millimetres in from the end, because the ends get chipped and worn. If you start measuring from the very end of the ruler, every length you read will be a little too big. Always put the **0 mark** on the starting point.",{"id":169,"type":170,"itemId":171,"prompt":172,"check":173,"hints":178,"feedback":180},"prac-read-ruler","practice","lines.discover-read-ruler","A pencil is placed along a ruler. One end is at the **2 cm** mark and the other end is at the **9.5 cm** mark. How long is the pencil, in cm?",{"kind":174,"answer":175,"tolerance":176,"unit":177},"number",7.5,0,"cm",[179],"The ruler did not start at 0, so subtract: bigger reading − smaller reading.",{"correct":181,"incorrect":182},"Yes: 9.5 − 2 = 7.5 cm.","Length = 9.5 − 2 = **7.5 cm**. When a length does not start at 0, subtract the two readings.",{"id":184,"type":170,"itemId":185,"prompt":186,"check":187,"hints":189,"feedback":191},"prac-count-square","lines.discover-count-square","Draw a square and then join two opposite corners with one straight line (a diagonal). How many line segments can you see in the drawing, counting each side and the diagonal once?",{"kind":174,"answer":188,"tolerance":176},5,[190],"A square has four sides.",{"correct":192,"incorrect":193},"Right: 4 sides + 1 diagonal = 5 segments.","Count the 4 sides of the square and add the 1 diagonal: **5** segments.",{"id":195,"type":135,"title":196,"problem":197,"steps":198},"we-count-triangle","How many segments hide in a triangle?","Draw a triangle and name its corners A, B and C. How many line segments are there?",[199,200,201],"Each side of the triangle joins two corners.","The sides are AB, BC and CA.","So there are **3** line segments. A square has 4, and a pentagon has 5: every flat shape with straight sides is made of segments.",{"id":203,"type":46,"variant":204,"title":205,"markdown":206},"mis-dot-segment","misconception","“A longer drawing means more points”","It is tempting to think a short segment has fewer points than a long one. In fact, **both** have endlessly many points. Between any two points on a segment, you can always find another point in the middle, and another between those, forever. Points have no size, so they never run out of room.",{"id":208,"type":57,"title":209,"eyebrow":210,"navLabel":211},"ch03","A ray: start here, go on forever","Chapter 03","3 Rays",{"id":213,"type":42,"markdown":214},"ray-torch","Switch on a torch in a dark room. The light starts at the bulb and shoots out in one direction. If nothing got in its way, where would it stop? It would not. It would go on and on, out of the window, past the Moon, into space.\n\nThat is the idea of a **ray**. A ray has a **starting point**, but it goes on **forever in one direction**. Sunbeams, laser pointers and the beam from a lighthouse are all good pictures of rays.",{"id":216,"type":46,"variant":87,"title":217,"markdown":218},"def-ray","Ray","A **ray** starts at one point, called its **end point** (or starting point), and goes on without end in one direction. We draw it with a dot at the start and an **arrowhead** at the other end to show that it keeps going. Ray AB starts at A and passes through B.",{"id":220,"type":66,"tone":221,"items":222},"spec-rays","copper",[223,227,231,235],{"label":224,"big":225,"value":226},"Torch beam","one start","Starts at the bulb and goes on in one direction.",{"label":228,"big":229,"value":230},"Sunbeam","from the Sun","Light leaves the Sun and travels outward in straight rays.",{"label":232,"big":233,"value":234},"Lighthouse","sweeping","The beam from the lamp is a ray that turns round and round.",{"label":236,"big":237,"value":238},"Clock hands","from the centre","Each hand is a picture of a ray starting at the centre of the clock.",{"id":240,"type":241,"prompt":242,"options":243,"explanation":256},"predict-ray-length","prediction","How long is a ray?",[244,247,250,253],{"id":245,"label":246},"a","As long as you draw it",{"id":248,"label":249},"b","Exactly as long as the arrow on the page",{"id":251,"label":252},"c","It has no length we can measure, because it never ends",{"id":254,"label":255},"d","Twice as long as a line segment","**It never ends, so we cannot give it a length.** The arrow you draw on paper is just a picture, like a road sign that says \"keeps going\". A line segment has a length because it has two ends. A ray has only one end, so there is nothing to measure *to*.",{"id":258,"type":46,"variant":259,"title":260,"markdown":261},"example-numberline","example","Rays and lines hiding in the number line","When you first learned counting numbers, you may have drawn them on a **ray**: start at 0 and mark 1, 2, 3, 4… going on forever to the right, with an arrow at the end. There is nothing to the left of 0 on that picture, so it has one end point. Later, when you meet negative numbers such as −1, −2 and −3, the picture becomes a **line**, with arrows at both ends, because the numbers go on forever both ways.",{"id":263,"type":46,"variant":264,"title":265,"markdown":266},"aha-ray-order","aha","The order of the letters matters for a ray","Ray AB starts at A and heads through B. Ray BA starts at B and heads through A. They point in **opposite directions**, so they are **different** rays. For a segment, AB and BA are the same thing, but for a ray the first letter always tells you where it starts.",{"id":268,"type":42,"markdown":269},"ray-many","Stand in an open field and point in any direction: north, east, towards a tree, towards a bird. Each direction you point in gives a different ray starting at you. How many directions are there? Endlessly many. So **endlessly many rays can start from the same point**.\n\nThink of the Sun drawn by a small child: a circle with rays sticking out in every direction. The child draws eight or twelve, but the real Sun sends light out in every direction at once.",{"id":271,"type":170,"itemId":272,"prompt":273,"check":274,"hints":286,"feedback":288},"prac-rays-one-point","lines.discover-rays-one-point","How many different rays can start from one point?",{"kind":275,"options":276,"correct":285},"choice",[277,279,281,283],{"id":245,"label":278},"Only one",{"id":248,"label":280},"Two, one each way",{"id":251,"label":282},"Four: up, down, left, right",{"id":254,"label":284},"Endlessly many",[254],[287],"Each direction you could point in gives a different ray.",{"correct":289,"incorrect":290},"Yes: every direction gives a new ray, and there are endlessly many directions.","Every direction you can point in from that point gives a different ray, so there are **endlessly many**.",{"id":292,"type":57,"title":293,"eyebrow":294,"navLabel":295},"ch04","A line: no start, no finish","Chapter 04","4 Lines",{"id":297,"type":42,"markdown":298},"line-intro","Take the segment AB and imagine stretching it **beyond both ends**, past A and past B, and never stopping in either direction. What you get is a **line**.\n\nA line goes on forever **both ways**. It has **no end points** at all. We draw a line with an **arrowhead at each end** to show that it keeps going. We can name it using any two points on it, as line AB, or with a small letter, as line *l*.",{"id":300,"type":46,"variant":87,"title":301,"markdown":302},"def-line","Line","A **line** is perfectly straight, has no thickness, and goes on without end in **both directions**. It has no end points and no length we can measure. Line AB and line BA are the same line.",{"id":304,"type":305,"caption":306,"columns":307,"rows":309},"table-three","table","The three straight things side by side",[308,128,217,301],"Feature",[310,314,319,323,328],[311,312,313,29],"End points","Two","One",[315,316,317,318],"How it is drawn","Dot at each end","Dot at one end, arrow at the other","Arrow at both ends",[320,321,322,322],"Length","Can be measured","Goes on forever",[324,325,326,327],"Name","segment AB (same as BA)","ray AB (not the same as BA)","line AB (same as BA), or line l",[329,330,331,332],"Everyday picture","Edge of a ruler, pencil, cricket pitch","Torch beam, sunbeam","An endless straight railway track",{"id":334,"type":335,"component":336,"componentVersion":5,"config":337,"objective":387,"textAlternative":388},"lab-sort-kinds","interactive","sort-game",{"prompt":338,"bins":339,"items":346,"seconds":176},"Is each thing best pictured as a line segment, a ray or a line?",[340,342,344],{"id":341,"label":128},"segment",{"id":343,"label":217},"ray",{"id":345,"label":301},"line",[347,351,355,359,363,367,371,375,379,383],{"id":348,"label":349,"bin":341,"why":350},"pencil","A new pencil","It has two ends and a length you can measure.",{"id":352,"label":353,"bin":341,"why":354},"pitch","A cricket pitch from stumps to stumps","It starts at one set of stumps and stops at the other: about 20.12 m.",{"id":356,"label":357,"bin":341,"why":358},"ruler-edge","The edge of a ruler","It stops at both ends of the ruler.",{"id":360,"label":361,"bin":343,"why":362},"torch","The beam of a torch","It starts at the bulb and goes on in one direction.",{"id":364,"label":365,"bin":343,"why":366},"sunbeam","A sunbeam through a window","Light leaves the Sun and travels outward in one direction.",{"id":368,"label":369,"bin":343,"why":370},"laser","A laser pointer's beam","It starts at the pointer and heads away in a straight line.",{"id":372,"label":373,"bin":343,"why":374},"lighthouse","A lighthouse beam at night","It starts at the lamp and shines out in one direction.",{"id":376,"label":377,"bin":345,"why":378},"numline","A number line with arrows at both ends","The arrows say the numbers go on forever both ways.",{"id":380,"label":381,"bin":345,"why":382},"endless-road","An imaginary straight road with no start or end","No end points at all: that is a line.",{"id":384,"label":385,"bin":341,"why":386},"rope","A rope stretched tight between two poles","It is tied at two ends, so it has two end points.","Sort everyday things into line segments, rays and lines.","This sorting game has ten everyday things. Ask two questions for each: does it have a start? Does it have a finish?\n\n**Line segments** (a start and a finish): a new pencil, a cricket pitch from stumps to stumps, the edge of a ruler, a rope tied tight between two poles.\n\n**Rays** (a start, no finish): a torch beam, a sunbeam, a laser pointer's beam, a lighthouse beam.\n\n**Lines** (no start, no finish): a number line with arrows at both ends, an imaginary straight road that never begins or ends.\n\nReal objects always end somewhere, so a real \"line\" is always an idea we imagine, not something we can hold.",{"id":390,"type":46,"variant":204,"title":391,"markdown":392},"mis-line-drawn","“The line on my page is 10 cm long”","What you drew is 10 cm long, but a mathematical line has no length. Your drawing is only a **piece** of the line, with arrows to remind you that the real line keeps going. That is why we can say two lines on a page will meet even if the drawings do not reach each other yet: we are allowed to imagine them continuing.",{"id":394,"type":335,"component":395,"componentVersion":5,"config":396,"objective":400,"textAlternative":401,"help":402},"lab-spot-kinds","line-spotter",{"modes":397,"rounds":399},[398],"kinds",6,"Spot whether each drawing is a line, a ray or a line segment by looking at its ends.","This game shows one straight drawing at a time, with two points labelled (for example A and B), and asks what it is: line AB, segment AB, ray AB or ray BA.\n\nThe trick is to look only at the **ends**:\n\n- A dot at **both** ends and no arrows: it stops at both ends, so it is a **line segment**.\n- A dot at **one** end and an arrowhead at the other: it starts somewhere and goes on forever one way, so it is a **ray**.\n- Arrowheads at **both** ends: it never stops either way, so it is a **line**.\n\nFor a ray, also check **where it starts**: if it starts at A and runs on through B, it is ray AB; if it starts at B, it is ray BA. The length of the drawing does not matter. After each answer the game explains why, and each correct answer scores a point and builds a streak.",{"simplerExplanation":403,"hints":404},"Count the arrows. Zero arrows: segment. One arrow: ray. Two arrows: line.",[405,406],"Ignore how long the drawing is. Only the ends matter.","A dot means \"it stops here\". An arrow means \"it keeps going\".",{"id":408,"type":170,"itemId":409,"prompt":410,"check":411,"hints":422,"feedback":424},"prac-arrows","lines.discover-arrows","A drawing has a dot at the left end and an arrowhead at the right end. What is it?",{"kind":275,"options":412,"correct":421},[413,415,417,419],{"id":245,"label":414},"A line segment",{"id":248,"label":416},"A ray",{"id":251,"label":418},"A line",{"id":254,"label":420},"A point",[248],[423],"How many ends does it stop at?",{"correct":425,"incorrect":426},"Yes: one end point, going on forever one way. That is a ray.","It stops at one end (the dot) and keeps going at the other (the arrow). That is a **ray**.",{"id":428,"type":57,"title":429,"eyebrow":430,"navLabel":431},"ch05","Straight or curved? Flat surfaces","Chapter 05","5 Curves and planes",{"id":433,"type":42,"markdown":434},"curve-intro","Not every path is straight. The outline of a bangle, the path of a thrown ball, the letter S and the edge of a leaf are **curves**. In everyday speech we sometimes say \"draw a curved line\", but in geometry the word **line** always means **straight**. A path that bends is called a **curve**.\n\nA curve that starts and ends at the same point, like a bangle or the letter O, is called a **closed curve**. One that does not, like the letter C, is an **open curve**.",{"id":436,"type":145,"title":437,"items":438},"steps-draw-three","How to draw each one neatly",[439,441,444,446],{"title":88,"text":440},"Make a small dot and write a capital letter next to it.",{"title":442,"text":443},"Segment","Mark two points, line up the ruler with both, and draw from one to the other. Stop exactly at the dots.",{"title":217,"text":445},"Mark the starting point and one more point. Draw from the start through the second point and a little beyond, then add an arrowhead.",{"title":301,"text":447},"Mark two points and draw through both, a little beyond each, with an arrowhead at both ends.",{"id":449,"type":305,"caption":450,"columns":451,"rows":454},"table-curves","Straight or curved?",[452,450,453],"Thing","Why",[455,459,463,467,471,475],[456,457,458],"Edge of a ruler","Straight","It never changes direction",[460,461,462],"A bangle","Curved (closed)","It bends all the way round and meets itself",[464,465,466],"A rainbow","Curved (open)","It bends and has two ends",[468,469,470],"A tight kite string","Almost straight","Pulling it tight removes the sag",[472,473,474],"A loose clothes line","Curved","It sags in the middle under its own weight",[476,477,478],"A railway track on a flat plain","Straight (over a short stretch)","Engineers lay it straight wherever they can",{"id":480,"type":42,"markdown":481},"plane-intro","Now think about a surface instead of a path. The top of a table, a sheet of paper, a blackboard, the floor of your classroom: they are all **flat**.\n\nImagine the top of your table growing wider and longer in every direction, forever, and staying perfectly flat and paper-thin. That is a **plane**. Every drawing you make in your notebook lives on a plane. Points, lines, rays and segments all sit on planes.",{"id":483,"type":46,"variant":87,"title":484,"markdown":485},"def-plane","Plane","A **plane** is a perfectly flat surface that goes on forever in every direction and has no thickness. A page, a table top and a floor are pictures of small pieces of a plane.",{"id":487,"type":46,"variant":488,"title":489,"markdown":490},"nuance-flat","nuance","Is the ground a plane?","A cricket ground looks flat, but it is part of the Earth, which is a ball. Over a few hundred metres the curve is far too small to notice, so treating the ground as a plane works perfectly. Over thousands of kilometres it does not, which is why pilots and map makers need different geometry. You will meet this idea again in the Extend layer.",{"id":492,"type":57,"title":493,"eyebrow":494,"navLabel":495},"ch06","When two lines cross","Chapter 06","6 Crossing lines",{"id":497,"type":42,"markdown":498},"intersect-intro","Open a pair of scissors. The two blades cross at the screw. Look at a road map: roads cross at junctions. Write the letter X: its two strokes cross in the middle.\n\nWhen two lines cross, we say they **intersect**. The place where they cross is a single point called the **point of intersection**. Two straight lines can cross in **only one point**: once they have crossed, they are heading away from each other and can never meet again.",{"id":500,"type":46,"variant":87,"title":501,"markdown":502},"def-intersecting","Intersecting lines","Two lines that meet at a point are **intersecting lines**. The point where they meet is the **point of intersection**. Two different straight lines can intersect in **at most one** point.",{"id":504,"type":66,"tone":505,"items":506},"spec-crossing","neutral",[507,511,515,519],{"label":508,"big":509,"value":510},"Scissors","✂","The blades cross at the screw, which is the point of intersection.",{"label":512,"big":513,"value":514},"Letter X","X","Two strokes crossing in the middle.",{"label":516,"big":517,"value":518},"Road junction","chowk","Two straight roads meeting at a crossing.",{"label":520,"big":521,"value":522},"Charpai rope","criss-cross","The woven ropes of a charpai cross each other hundreds of times.",{"id":524,"type":241,"prompt":525,"options":526,"explanation":533},"predict-two-cross","Two straight lines cross once at point P. Can they cross again somewhere else?",[527,529,531],{"id":245,"label":528},"Yes, if you make them long enough",{"id":248,"label":530},"Yes, but only once more",{"id":251,"label":532},"No, two straight lines can meet at most once","**No.** After crossing at P, the lines are moving apart. Being straight, they cannot turn back towards each other. If two \"lines\" seem to meet twice, at least one of them must be curved.",{"id":535,"type":57,"title":536,"eyebrow":537,"navLabel":538},"ch07","Lines that never meet","Chapter 07","7 Parallel lines",{"id":540,"type":42,"markdown":541},"parallel-intro","Look at the lines printed in your notebook. However far along the page you go, they never get closer together and never get further apart. Now picture a railway track: two steel rails, always the same distance apart, running for hundreds of kilometres. If the rails ever came together, the train would fall off!\n\nLines like these are called **parallel lines**. They lie on the same flat surface, they stay the **same distance apart** all the way along, and so they **never meet**, however far you extend them.",{"id":543,"type":46,"variant":87,"title":544,"markdown":545},"def-parallel","Parallel lines","Two lines on the same flat surface (plane) that **never meet**, however far they are extended, are **parallel lines**. They stay the same distance apart everywhere. We write \"line AB is parallel to line CD\" as AB ∥ CD.",{"id":547,"type":66,"tone":67,"items":548},"spec-parallel",[549,553,557,561,565],{"label":550,"big":551,"value":552},"Railway rails","1,676 mm","India's broad-gauge rails are kept exactly this distance apart.",{"label":554,"big":555,"value":556},"Notebook lines","ruled","Printed the same distance apart all down the page.",{"label":558,"big":559,"value":560},"Zebra crossing","stripes","The white stripes are parallel to one another.",{"label":562,"big":563,"value":564},"Ladder","two rails","The two long side rails of a ladder are parallel.",{"label":566,"big":567,"value":568},"Harmonium keys","side by side","The edges of neighbouring keys are parallel.",{"id":570,"type":46,"variant":264,"title":571,"markdown":572},"aha-tracks","Why do railway tracks seem to meet?","Stand on a long straight platform and look down the line. The two rails seem to come closer and closer until they meet at a point near the horizon. They do not really meet. Things further away look smaller, so the gap between the rails *looks* smaller too. Artists use this trick to make drawings look deep; you will explore it in the Extend layer.",{"id":574,"type":57,"title":575,"eyebrow":576,"navLabel":577},"ch08","Square corners: perpendicular lines","Chapter 08","8 Perpendicular",{"id":579,"type":42,"markdown":580},"perp-intro","Look at the corner of a book, a door frame, a window, or the plus sign +. In each one, two straight edges meet to make a perfect **square corner**. That square corner is called a **right angle**, and it measures 90 degrees (written 90°).\n\nTwo lines that cross or meet at a right angle are called **perpendicular lines**. They are a special kind of intersecting line: they do not just meet, they meet \"square\".",{"id":582,"type":46,"variant":87,"title":583,"markdown":584},"def-perp","Perpendicular lines","Two lines that meet at a **right angle** (90°) are **perpendicular**. We write \"line AB is perpendicular to line CD\" as AB ⊥ CD. The symbol ⊥ is itself a picture of two perpendicular lines.",{"id":586,"type":46,"variant":95,"title":587,"markdown":588},"tryit-fold","Make a right-angle checker from any paper","Take a scrap of paper with a rough, torn edge. Fold it once anywhere: the fold is a straight line. Now fold it again so that the first fold lies exactly on top of itself. Open it out. The two creases are **perpendicular**, and the folded corner is a perfect right angle. Use it to test the corners of books, tiles and doors around your home. Which corners are really square?",{"id":590,"type":305,"caption":591,"columns":592,"rows":597},"table-kinds-pairs","Three ways two lines on a page can sit",[593,594,595,596],"Relationship","Do they meet?","What it looks like","Everyday example",[598,603,608],[599,600,601,602],"Intersecting","Yes, at one point","Crossing, any angle","Scissors, letter X, road junction",[604,605,606,607],"Perpendicular","Yes, at one point, at 90°","A square corner or a plus sign","Window frame, the letter T, a flag on its pole",[609,610,611,612],"Parallel","Never","Side by side, same gap everywhere","Railway rails, notebook lines, zebra stripes",{"id":614,"type":335,"component":395,"componentVersion":5,"config":615,"objective":618,"textAlternative":619,"help":620},"lab-spot-pairs",{"modes":616,"rounds":399},[617],"pairs","Decide whether each pair of lines is parallel, perpendicular or just intersecting.","This game shows two straight lines at a time. Decide how they are related:\n\n- **Parallel:** the gap between them is the same everywhere and they will never meet, even if you imagine them going on forever.\n- **Perpendicular:** they meet at a square corner (a right angle, 90°), like a plus sign or the letter T.\n- **Intersecting:** they cross at some other angle, like the letter X leaning over.\n\nThe lines are drawn right across the picture and can be turned at any angle, so a perpendicular pair may look like a tilted plus sign. Sometimes a small square is drawn in the corner to mark a right angle. Every right-angle pair is also intersecting, but the game wants the most exact name: perpendicular.",{"simplerExplanation":621,"hints":622},"Same gap all along: parallel. Square corner: perpendicular. Any other crossing: intersecting.",[623,624],"Imagine the lines going on forever. Would they ever bump into each other?","Fit the corner of a book into the crossing. Does it fit exactly?",{"id":626,"type":241,"prompt":627,"options":628,"explanation":635},"predict-rails","You stand on a long, straight railway platform and look along the track. What do the two rails seem to do far away?",[629,631,633],{"id":245,"label":630},"Get further apart",{"id":248,"label":632},"Seem to come together at a point",{"id":251,"label":634},"Look exactly the same distance apart all the way","They **seem to come together** at a point near the horizon, although they are really parallel and always the same distance apart. Far-away things look smaller, and that includes the gap between the rails.",{"id":637,"type":170,"itemId":638,"prompt":639,"check":640,"hints":647,"feedback":649},"prac-plus","lines.discover-plus-sign","The two strokes of a plus sign + are:",{"kind":275,"options":641,"correct":646},[642,643,644],{"id":245,"label":609},{"id":248,"label":604},{"id":251,"label":645},"Neither: they do not meet",[248],[648],"Do the strokes meet? At what kind of corner?",{"correct":650,"incorrect":651},"Right: they cross at four square corners, so they are perpendicular.","The strokes cross at square corners (right angles), so they are **perpendicular**.",{"id":653,"type":57,"title":654,"eyebrow":655,"navLabel":656},"ch09","Standing up and lying down","Chapter 09","9 Horizontal, vertical",{"id":658,"type":42,"markdown":659},"hv-intro","Look at the edge of a still pond or the surface of water in a glass. It lies perfectly flat and level. A line that is level like this is called **horizontal** (from the word *horizon*, the line where the sky seems to meet the land or the sea).\n\nNow hang a small stone on a thread and hold the thread still. The thread points straight down towards the centre of the Earth. A line that points straight up and down like this is called **vertical**. A horizontal line and a vertical line always meet at a right angle, so they are perpendicular.",{"id":661,"type":66,"tone":505,"items":662},"spec-hv",[663,667,671,675],{"label":664,"big":665,"value":666},"Plumb line","straight down","A weight on a string. Masons use it to check that a wall is vertical.",{"label":668,"big":669,"value":670},"Spirit level","bubble","A tube of liquid with a bubble. When the bubble is centred, the surface is horizontal.",{"label":672,"big":673,"value":674},"Horizon","sea and sky","Out at sea, the horizon looks like a horizontal line.",{"label":676,"big":677,"value":678},"Flag pole","upright","A flag pole should stand vertical, perpendicular to level ground.",{"id":680,"type":46,"variant":204,"title":681,"markdown":682},"mis-perp-hv","“Perpendicular means one line is standing up”","Turn this page on its side and the corner of a book is still a square corner. Perpendicular only means the two lines meet at **90°**. The lines can be tilted any way at all. A horizontal line and a vertical line are always perpendicular, but perpendicular lines do not have to be horizontal and vertical.",{"id":684,"type":170,"itemId":685,"prompt":686,"check":687,"hints":696,"feedback":698},"prac-tilted","lines.discover-tilted-perp","A tilted square tile has two edges meeting at its corner. Neither edge is horizontal. Are they perpendicular?",{"kind":275,"options":688,"correct":695},[689,691,693],{"id":245,"label":690},"Yes: the corner is still a right angle",{"id":248,"label":692},"No: one edge must be vertical",{"id":251,"label":694},"No: tilted lines can only be parallel",[245],[697],"Does tilting a square change its corners?",{"correct":699,"incorrect":700},"Right: turning a shape does not change its angles.","Turning a tile does not change its corners. They are still right angles, so the edges are **perpendicular**.",{"id":702,"type":57,"title":703,"eyebrow":704,"navLabel":705},"ch10","Lines all around you","Chapter 10","10 Lines around you",{"id":707,"type":42,"markdown":708},"around-intro","Now that you know the names, you will start seeing them everywhere. Games are a good place to look, because courts and pitches are painted with lines that follow strict rules.",{"id":710,"type":42,"markdown":711},"letters-intro","Even the letters you write are built from lines. Capital letters in a plain font are a great place to practise spotting parallel and perpendicular strokes.",{"id":713,"type":305,"caption":714,"columns":715,"rows":720},"table-letters","Capital letters made of straight strokes",[716,717,718,719],"Letter","Parallel strokes?","Perpendicular strokes?","Other crossings",[721,725,729,733,736,740,743,746],[722,723,724,29],"H","Yes: the two uprights","Yes: the bar meets each upright at 90°",[726,727,728,29],"E","Yes: the three arms","Yes: each arm meets the upright at 90°",[730,731,732,29],"T","No","Yes: the top bar and the stem",[734,731,735,29],"L","Yes",[737,738,731,739],"Z","Yes: top and bottom","The slanted middle meets them at a slant",[741,723,731,742],"N","The slant meets each upright at a slant",[513,731,744,745],"Not usually","The two strokes cross at a slant",[747,731,731,748],"V","Two strokes meet at a point at the bottom",{"id":750,"type":170,"itemId":751,"prompt":752,"check":753,"hints":755,"feedback":758},"prac-letters","lines.discover-letters","In a plain capital-letter font, how many of the letters in the word **FLAT** have at least one pair of perpendicular strokes?",{"kind":174,"answer":754,"tolerance":176},3,[756,757],"Check F, L, A and T one at a time.","A is made of two slanted strokes and a cross-bar. Do any meet at a square corner?",{"correct":759,"incorrect":760},"Yes: F, L and T do; A does not.","F (arms meet the upright at 90°), L and T all have perpendicular strokes. A's strokes meet at slants. So the answer is **3**.",{"id":762,"type":763,"title":764,"prompt":765,"options":766},"explorer-around","explorer","Where lines hide in everyday life","Pick a place to see which kinds of lines are hiding in it.",[767,778,789,799,809],{"id":768,"label":115,"chain":769,"badge":774,"note":777},"cricket",[770,771,772,773],"Bowling crease","Popping crease parallel to it","Return creases perpendicular","Stumps 22 yards apart",{"text":775,"tone":776},"Parallel and perpendicular","yes","At each end of the pitch, the **popping crease** is painted parallel to the **bowling crease**, 1.22 m (4 feet) in front of it. The **return creases** run at right angles to them, so they are perpendicular. The batter is safe when the bat is grounded behind the popping crease.",{"id":779,"label":780,"chain":781,"badge":786,"note":788},"badminton","Badminton court",[782,783,784,785],"Side lines parallel","Back lines parallel","Every corner a right angle","Net line across the middle",{"text":787,"tone":776},"A rectangle of lines","A badminton court is 13.4 m long. The two long side lines are parallel, the two back lines are parallel, and every place where a side line meets a back line is a right angle, so those pairs are perpendicular. The short service lines are parallel to the net.",{"id":790,"label":791,"chain":792,"badge":796,"note":798},"railway","Railway track",[793,794,609,795],"Two rails","Always 1,676 mm apart","Sleepers cross them",{"text":797,"tone":776},"Parallel rails","On India's broad gauge the rails are kept 1,676 mm apart. The concrete **sleepers** under them run across both rails, roughly perpendicular to them, and hold the gap steady.",{"id":800,"label":801,"chain":802,"badge":806,"note":808},"road","City roads",[803,804,805,558],"Straight roads","Crossing at junctions","Lane markings parallel",{"text":807,"tone":776},"Intersecting and parallel","Roads crossing at a junction are intersecting lines, and in a planned city like Chandigarh many cross at right angles. The painted lane lines along one road are parallel, and so are the stripes of a zebra crossing.",{"id":810,"label":811,"chain":812,"badge":817,"note":819},"wires","Electric wires",[813,814,815,816],"Pole to pole","Wires stretched side by side","Almost parallel","Sag in the middle",{"text":815,"tone":818},"no","Overhead wires between two poles run side by side and look parallel. But each one **sags** a little under its own weight, so really each wire is a gentle curve, not a straight line. Mathematics gives a perfect picture; the real world is always a little rough.",{"id":821,"type":335,"component":336,"componentVersion":5,"config":822,"objective":881,"textAlternative":882},"lab-sort-everyday",{"prompt":823,"bins":824,"items":832,"seconds":176},"Sort each everyday thing by how its lines sit.",[825,827,829],{"id":826,"label":609},"parallel",{"id":828,"label":604},"perpendicular",{"id":830,"label":831},"intersecting","Intersecting (not square)",[833,837,841,845,849,853,857,861,865,869,873,877],{"id":834,"label":835,"bin":826,"why":836},"rails","The two rails of a railway track","They stay the same distance apart and never meet.",{"id":838,"label":839,"bin":826,"why":840},"notebook","Printed lines in a notebook","Equal gaps all down the page.",{"id":842,"label":843,"bin":826,"why":844},"zebra","Stripes of a zebra crossing","Each stripe runs alongside the next at the same gap.",{"id":846,"label":847,"bin":826,"why":848},"ladder","The two long sides of a ladder","The rungs stay the same length, so the sides stay the same distance apart.",{"id":850,"label":851,"bin":828,"why":852},"plus","The two strokes of a plus sign","They cross at square corners.",{"id":854,"label":855,"bin":828,"why":856},"window","Top and side of a window frame","Window corners are right angles.",{"id":858,"label":859,"bin":828,"why":860},"flag","A flag pole and the ground","A pole stands straight up, at a right angle to level ground.",{"id":862,"label":863,"bin":828,"why":864},"book","Two edges at the corner of a book","A book corner is a square corner.",{"id":866,"label":867,"bin":830,"why":868},"scissors","Open scissor blades","They cross at the screw but not at a square corner.",{"id":870,"label":871,"bin":830,"why":872},"x","The strokes of the letter X","They cross in the middle at a slant.",{"id":874,"label":875,"bin":830,"why":876},"roof","The two slopes of a roof seen from the front","They meet at the top at a pointed angle.",{"id":878,"label":879,"bin":828,"why":880},"kite-sticks","The two sticks of a paper kite","Kite sticks are tied at right angles to each other.","Sort everyday objects into parallel, perpendicular and intersecting (not at a right angle).","This sorting game gives twelve everyday things. For each, decide how its two straight lines sit.\n\n**Parallel:** railway rails, notebook lines, zebra-crossing stripes, the two long sides of a ladder. They never meet and keep the same gap.\n\n**Perpendicular:** a plus sign, the top and side of a window frame, a flag pole and flat ground, the corner of a book, the two sticks of a paper kite. They meet at square corners.\n\n**Intersecting but not square:** open scissor blades, the letter X, the two slopes of a roof. They cross at a slant.\n\nRemember that perpendicular lines are a special kind of intersecting line. Here, put a pair in the *most exact* bin.",{"id":884,"type":57,"title":885,"eyebrow":886,"navLabel":887},"ch11","Lines painted on grounds and roads","Chapter 11","11 Courts and roads",{"id":889,"type":42,"markdown":890},"courts-intro","Walk onto any sports ground in a school in India and you are standing on a giant geometry diagram. The lines are painted so that referees can make fair decisions: is the player in or out? Did the raider cross the line? Every one of those decisions depends on knowing exactly where a straight line is.",{"id":892,"type":66,"tone":108,"items":893},"spec-kabaddi",[894,898,902,906,910],{"label":895,"big":896,"value":897},"Kabaddi court","13 m × 10 m","The size for men under the current international rules; women's and junior courts are smaller.",{"label":899,"big":900,"value":901},"Mid line","across the middle","Divides the court into two halves, one for each team.",{"label":903,"big":904,"value":905},"Baulk line","3.75 m from mid line","A raider must cross this line in the opponents' half. It is parallel to the mid line.",{"label":907,"big":908,"value":909},"Bonus line","1 m beyond baulk line","Touching it can earn a bonus point. It is parallel to the baulk line too.",{"label":911,"big":912,"value":913},"Side lines","the long edges","Perpendicular to the mid, baulk and bonus lines.",{"id":915,"type":135,"title":916,"problem":917,"steps":918},"we-kabaddi","How far is the bonus line from the mid line?","On a kabaddi court the baulk line is 3.75 m from the mid line, and the bonus line is 1 m further on. How far is the bonus line from the mid line? What is the relationship between the three lines?",[919,920,921],"The mid line, baulk line and bonus line all run across the court, the same direction, and never meet: they are **parallel**.","Distance from the mid line to the bonus line = 3.75 m + 1 m = **4.75 m**.","Because the lines are parallel, this distance is the same wherever you measure it across the court, as long as you measure straight (perpendicular to the lines).",{"id":923,"type":170,"itemId":924,"prompt":925,"check":926,"hints":930,"feedback":933},"prac-kabaddi-width","lines.discover-kabaddi-half","A kabaddi court is 13 m long, and the mid line cuts it into two equal halves. How long is each half, in metres?",{"kind":174,"answer":927,"tolerance":928,"unit":929},6.5,0.001,"m",[931,932],"The mid line is halfway along.","Divide the length by 2.",{"correct":934,"incorrect":935},"Yes: 13 ÷ 2 = 6.5 m.","The mid line is in the middle, so each half is 13 ÷ 2 = **6.5 m** long.",{"id":842,"type":42,"markdown":937},"Now look at a **zebra crossing**. Each white stripe is a long, thin rectangle. On a standard crossing the stripes are **parallel** to one another, and the long side of each one runs along the road, the way the traffic moves; they are laid side by side across the road. The path you walk along to cross the road runs **perpendicular** to the road's edge: straight across, the shortest way to the other side. That is not an accident. The perpendicular path is the shortest one, so you spend the least time in the road.",{"id":939,"type":46,"variant":204,"title":940,"markdown":941},"mis-zebra","\\u201cThe stripes point across the road\\u201d","Look carefully next time: the long sides of the zebra stripes run **along** the road, the same way as the cars, not across it. It is the crossing as a whole, and the path you walk, that goes across. Many children draw the stripes the wrong way round.",{"id":943,"type":42,"markdown":944},"wires-home","Even the walls of your home hide lines. Electricians usually run the wires inside a wall in straight **vertical** and **horizontal** paths: straight up or down from a switch board, and level along the wall near the ceiling. That way, anyone who later hammers a nail into the wall knows where the wires are likely to be: never directly above or below a switch.",{"id":946,"type":46,"variant":165,"title":947,"markdown":948},"careful-wires","Never drill near a switch board","Because wires often run vertically above and below switches and sockets, never hammer nails or drill into a wall in a straight line above or below a switch board. Always ask an adult, and let them check first. Never touch electric wires yourself.",{"id":950,"type":170,"itemId":951,"prompt":952,"check":953,"hints":963,"feedback":965},"prac-wall-wires","lines.discover-wall-wires","A switch board is on a wall. The wire from it goes straight up to the ceiling. Which kind of line does the wire follow?",{"kind":275,"options":954,"correct":962},[955,957,959,961],{"id":245,"label":956},"Horizontal",{"id":248,"label":958},"Vertical",{"id":251,"label":960},"A curve",{"id":254,"label":416},[248],[964],"Straight up and down like a plumb line.",{"correct":966,"incorrect":967},"Yes: straight up is vertical.","A path straight up and down is **vertical**, like a thread with a stone hanging from it.",{"id":969,"type":170,"itemId":970,"prompt":971,"check":972,"hints":980,"feedback":982},"prac-court-lines","lines.discover-court-lines","On a badminton court, the two long side lines never meet and are always the same distance apart. They are:",{"kind":275,"options":973,"correct":979},[974,975,976,977],{"id":245,"label":604},{"id":248,"label":609},{"id":251,"label":599},{"id":254,"label":978},"Rays",[248],[981],"Do they ever meet?",{"correct":983,"incorrect":984},"Right: parallel.","Lines that never meet and stay the same distance apart are **parallel**.",{"id":986,"type":46,"variant":204,"title":987,"markdown":988},"mis-seg-line-d","\\u201cA side line is a line, so it goes on forever\\u201d","In everyday speech we call court markings \\\"lines\\\", but in geometry each painted side line is a **line segment**: it starts at one corner of the court and stops at another, and you can measure it. Mathematicians would say the side line is a segment that lies *on* a line. Keep the everyday word and the maths word apart in your head.",{"id":990,"type":57,"title":991,"eyebrow":992,"navLabel":993},"ch12","Putting it together","Chapter 12","12 Wrap-up",{"id":995,"type":335,"component":996,"componentVersion":5,"config":997,"objective":1017,"textAlternative":1018},"lab-match-words","match-pairs",{"prompt":998,"mode":999,"pairs":1000},"Match each word to its picture in words.","memory",[1001,1003,1005,1006,1008,1010,1012,1014],{"a":88,"b":1002},"An exact place, shown by a dot",{"a":128,"b":1004},"Dot at both ends",{"a":217,"b":317},{"a":301,"b":1007},"Arrows at both ends",{"a":544,"b":1009},"Never meet, same gap everywhere",{"a":583,"b":1011},"Meet at a square corner",{"a":484,"b":1013},"A flat surface that goes on forever",{"a":1015,"b":1016},"Curve","A path that bends","Match each geometry word with a short description of what it looks like.","This is a memory game with eight pairs of cards. Turn over two cards; keep them if they match.\n\nThe pairs are: **point**, an exact place shown by a dot; **line segment**, dot at both ends; **ray**, dot at one end and an arrow at the other; **line**, arrows at both ends; **parallel lines**, never meet and keep the same gap; **perpendicular lines**, meet at a square corner; **plane**, a flat surface that goes on forever; **curve**, a path that bends.",{"id":1020,"type":46,"variant":95,"title":1021,"markdown":1022},"tryit-hunt","A five-minute line hunt","Set a timer for five minutes and walk around your home. Make three columns in your notebook: **parallel**, **perpendicular** and **intersecting (slanted)**. Write down everything you find. Then compare with a friend or family member. Who found the most? Which column was longest? Most homes are full of perpendicular lines, because doors, windows, tiles and shelves are built with square corners.",{"id":1024,"type":1025,"title":1026,"terms":1027},"glossary-discover","glossary","Words from this lesson",[1028,1031,1034,1038,1041,1044,1047,1050,1053,1057,1059,1062],{"term":88,"meaning":1029,"example":1030},"An exact position. It has no size. Named with a capital letter.","Point A",{"term":128,"meaning":1032,"example":1033},"The straight path between two points. It has two end points and a length.","Segment AB, 6 cm long",{"term":1035,"meaning":1036,"example":1037},"End point","A point where a segment or ray stops.","A and B are the end points of segment AB",{"term":217,"meaning":1039,"example":1040},"Part of a line that starts at one point and goes on forever in one direction.","A torch beam",{"term":301,"meaning":1042,"example":1043},"A straight path with no end points that goes on forever in both directions.","Line AB, or line l",{"term":1015,"meaning":1045,"example":1046},"A path that bends. In geometry, a line is always straight; a bending path is a curve.","The outline of a bangle",{"term":484,"meaning":1048,"example":1049},"A perfectly flat surface that goes on forever in every direction.","A table top is a small piece of a plane",{"term":501,"meaning":1051,"example":1052},"Lines that meet or cross at a point.","The blades of open scissors",{"term":1054,"meaning":1055,"example":1056},"Point of intersection","The single point where two lines cross.","The screw of a pair of scissors",{"term":544,"meaning":1058,"example":550},"Lines on the same plane that never meet and stay the same distance apart.",{"term":583,"meaning":1060,"example":1061},"Lines that meet at a right angle (90°).","The strokes of a plus sign",{"term":1063,"meaning":1064,"example":1065},"Right angle","A square corner, measuring 90°.","The corner of a book",{"id":1067,"type":1068,"title":1069,"questions":1070},"quiz-discover","quiz","Check your eye for lines",[1071,1084,1094,1104,1117,1127,1136,1147,1158],{"itemId":1072,"prompt":1073,"options":1074,"correct":248,"why":1083},"lines.discover-q-point","What does a point show?",[1075,1077,1079,1081],{"id":245,"label":1076},"A small circle",{"id":248,"label":1078},"An exact position",{"id":251,"label":1080},"A very short line",{"id":254,"label":1082},"A length of 1 mm","A point marks an exact place and has no size. The dot is only a picture of it.",{"itemId":1085,"prompt":1086,"options":1087,"correct":251,"why":1093},"lines.discover-q-segment","Which has exactly two end points?",[1088,1089,1090,1091],{"id":245,"label":418},{"id":248,"label":416},{"id":251,"label":414},{"id":254,"label":1092},"A plane","A line segment stops at both ends, so it has two end points and a length.",{"itemId":1095,"prompt":1096,"options":1097,"correct":248,"why":1103},"lines.discover-q-ray","Which is the best everyday picture of a ray?",[1098,1099,1100,1101],{"id":245,"label":111},{"id":248,"label":1040},{"id":251,"label":460},{"id":254,"label":1102},"A table top","A torch beam starts at the bulb and goes on in one direction.",{"itemId":1105,"prompt":1106,"options":1107,"correct":245,"why":1116},"lines.discover-q-line","How many end points does a line have?",[1108,1110,1112,1114],{"id":245,"label":1109},"0",{"id":248,"label":1111},"1",{"id":251,"label":1113},"2",{"id":254,"label":1115},"It depends on how long it is","A line goes on forever both ways, so it has no end points.",{"itemId":1118,"prompt":1119,"options":1120,"correct":251,"why":1126},"lines.discover-q-rails","Railway rails are a picture of…",[1121,1122,1123,1124],{"id":245,"label":501},{"id":248,"label":583},{"id":251,"label":544},{"id":254,"label":1125},"Curves only","The rails stay the same distance apart and never meet.",{"itemId":1128,"prompt":1129,"options":1130,"correct":248,"why":1135},"lines.discover-q-corner","The corner of a book shows two edges that are…",[1131,1132,1133,1134],{"id":245,"label":609},{"id":248,"label":604},{"id":251,"label":473},{"id":254,"label":978},"They meet at a right angle, a square corner, so they are perpendicular.",{"itemId":1137,"prompt":1138,"options":1139,"correct":245,"why":1146},"lines.discover-q-cross","Two straight lines cross. How many points of intersection can they have?",[1140,1141,1142,1144],{"id":245,"label":313},{"id":248,"label":312},{"id":251,"label":1143},"As many as you like",{"id":254,"label":1145},"None, ever","Two different straight lines can meet at most once.",{"itemId":1148,"prompt":1149,"options":1150,"correct":248,"why":1157},"lines.discover-q-curve","Which of these is a curve, not a line?",[1151,1152,1154,1155],{"id":245,"label":357},{"id":248,"label":1153},"The path of a thrown ball",{"id":251,"label":468},{"id":254,"label":1156},"A notebook line","A thrown ball rises and falls along a bending path.",{"itemId":1159,"prompt":1160,"options":1161,"correct":248,"why":1170},"lines.discover-q-name","How do we usually name a point?",[1162,1164,1166,1168],{"id":245,"label":1163},"With a number",{"id":248,"label":1165},"With a capital letter",{"id":251,"label":1167},"With a small letter",{"id":254,"label":1169},"With a colour","Points are named with capital letters, such as A, B or P.",{"id":1172,"type":1173,"prompt":1174},"reflect-discover","reflection","Walk around your home or school and find one example each of a line segment, parallel lines and perpendicular lines. Which one was hardest to find, and why?",{"id":1176,"type":1177,"title":1178,"points":1179},"cheat-discover","summary","Cheat sheet",[1180,1181,1182,1183,1184,1185,1186,1187,1188],"**Point:** an exact place, no size. Draw a dot; name it with a capital letter.","**Line segment:** straight path between two points. Two end points, a length you can measure. AB = BA.","**Ray:** one end point, goes on forever one way. Ray AB starts at A; ray BA starts at B.","**Line:** no end points, goes on forever both ways. Arrows at both ends.","**Curve:** a path that bends. In geometry, \"line\" always means straight.","**Plane:** a flat surface that goes on forever, like an endless table top.","**Intersecting lines** cross at exactly one point, the point of intersection.","**Parallel lines** never meet and keep the same gap: railway rails, notebook lines.","**Perpendicular lines** meet at a right angle (90°): a plus sign, a book corner.",{"id":1190,"type":1191,"conceptId":1192,"relation":1193,"explanation":1194},"conn-angles","connection","angles","helps_understand","Two rays that start at the same point make an angle, so rays are the building blocks of every angle.",{"id":1196,"type":1191,"conceptId":1197,"relation":1193,"explanation":1198},"conn-shape","shape-and-space","Every flat shape is built from line segments: a triangle from three, a square from four.",{"id":1200,"type":1191,"conceptId":1201,"relation":1202,"explanation":1203},"conn-patterns","patterns","related_to","Matchstick patterns are made of equal line segments, and counting them leads to number patterns.",{"id":1205,"type":1206,"sourceIds":1207},"sources-discover","sources",[1208,1209,1210,1211,1212,1213,1214,1215,1216],"lines-ncert-math-6","lines-ncert-ganita-prakash-6","lines-mathsisfun-line","lines-mathsisfun-parallel-perpendicular","lines-mcc-laws-creases","lines-wiki-kabaddi","lines-wkf-rules-kabaddi","lines-irc-35-road-markings","lines-wiki-rail-transport-india",[1208,1209,1210,1211,1212,1213,1214,1215,1216],"needs_review",{"generatedBy":1220,"notes":1221},"claude-code","Draft generated by a scripted generator; every count and length was computed in Python. Pending owner review.","eac0c410621cc363eaed3103ccc6e82cefd6fc815f61b2b1eda3188c6eee17f2",{"logic:practice":1224,"component:sort-game@1":1225,"component:line-spotter@1":1226,"component:match-pairs@1":1227,"source:lines-irc-35-road-markings":1228,"source:lines-mathsisfun-line":1229,"source:lines-mathsisfun-parallel-perpendicular":1230,"source:lines-mcc-laws-creases":1231,"source:lines-ncert-ganita-prakash-6":1232,"source:lines-ncert-math-6":1233,"source:lines-wiki-kabaddi":1234,"source:lines-wiki-rail-transport-india":1235,"source:lines-wkf-rules-kabaddi":1236},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","1279a4e23634aba0fd1cb2bb76638a44d6b592131c326e194fdd1bf593a12f93","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","f7987084cd045541cbd20cd4782d1f80d4737eabbdde880e6e812b7dacdff140","982f31ecb1e1e17e74ba13872e9e63e218010de6c5653901ebd0db8a12413bc3","ee8cac463d4577f5a05460d731e8e92e3d4268172d2221efc74a4e61e4376345","29053ec2c573f86831c323ce2ecab04945b506bf04bddd43c7582c9fbf98ce8b","b7ec4502e5c7912ecccfeb371ab30a19043c5598016c6d51acbc678290fc40a2","219993d3ebc010eeac3a16481eda537da1557ef707d475cbca1ba05a49dfff80","c3c59ba667c814a640c01ea119d156347dc04c3844e0067f4e60f9ffae877e8f","4c30199ad6fdba1851b0f0e0521ee8b521206bf1d70413dad7404cb0cedbb3e3","36a88d8b6dff9f0081a9fe40a5638ea5c9411372182ccf338b1a59568efe35d3",{"state":1238,"reviewer":1239,"selfReview":1240,"reviewedAt":1241,"method":1242},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598773]