[{"data":1,"prerenderedAt":1097},["ShallowReactive",2],{"layer:shape-and-space:investigate":3},{"layer":4,"contentHash":1075,"dependencyHashes":1076,"approval":1090,"releaseId":1096},{"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":1070,"reviewStatus":1071,"authoring":1072},1,"shape-and-space","en","investigate","Test it, fold it, count it","Predictions and experiments with diagonals, triangles, nets, views, symmetry and π","Predict, then test: how fast diagonals multiply, which three sticks make a triangle, what polygon angles add up to, which statements are always true, the F + V − E pattern, which six-square shapes fold into a cube, symmetry in letters, measuring π and which shapes tile a floor.",[13,14,15,16,17],"Find and extend patterns in the diagonals and angle sums of polygons.","Test the triangle inequality with real lengths and explain the flat (degenerate) case.","Decide whether statements about shapes are always, sometimes or never true, with examples.","Discover F + V − E = 2 for polyhedra and test which six-square arrangements fold into a cube.","Measure circumference ÷ diameter, and explain which regular polygons tile a floor and why.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 45 minutes plus experiments",{"label":29,"value":30},"Prior knowledge","Understand layer",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","6: sorts, F\u002FE\u002FV hunt, views match, tiling",{"label":38,"value":39},"Kit","Straws, squared paper, thread, mirror",[41,45,51,57,75,122,127,142,147,150,161,198,203,224,230,235,239,252,283,287,300,305,310,313,396,401,405,410,413,438,469,473,484,489,492,559,563,573,578,581,591,623,628,633,637,712,721,726,729,763,767,780,785,790,793,810,823,844,849,852,869,873,909,1029,1033,1049,1053,1057],{"id":42,"type":43,"markdown":44},"intro-investigate","prose","In this layer you stop being told and start **finding out**. Every chapter begins with a question you can test with paper, straws, a thread, a box or a lab. Before each experiment you will make a **prediction**. Being wrong is not a failure here: it is the moment you learn something.\n\nMathematicians work exactly like this. They try examples, spot a pattern, guess a rule, then hunt for an example that breaks it. Only when the rule survives every test do they try to prove it (which is what the next layer, Deepen, is about).",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-kit","callout","try_it","Your investigation kit","Collect: 10–15 drinking straws or broom sticks, scissors, a ruler, sticky tape, a thread or string, several sheets of paper (squared paper if you have it), a few small boxes, some round objects (bangle, plate, tumbler) and a small mirror. Keep a notebook for your predictions and results.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","How fast do diagonals multiply?","Chapter 01","1 Diagonal pattern",{"id":58,"type":59,"prompt":60,"options":61,"explanation":74},"pred-decagon","prediction","A triangle has 0 diagonals, a quadrilateral 2, a pentagon 5, a hexagon 9. Predict how many diagonals a **decagon** (10 sides) has.",[62,65,68,71],{"id":63,"label":64},"a","About 15",{"id":66,"label":67},"b","About 20",{"id":69,"label":70},"c","About 35",{"id":72,"label":73},"d","About 50","**35.** The numbers grow faster than you might expect. From each of the 10 vertices, 7 diagonals leave (all vertices except itself and its 2 neighbours). That is 10 × 7 = 70 ends, and each diagonal has 2 ends, so 70 ÷ 2 = 35. Look at the table below to see how the jumps between polygons grow by one each time.",{"id":76,"type":77,"caption":78,"columns":79,"rows":84},"table-diag-growth","table","Diagonals as the number of sides grows",[80,81,82,83],"Sides","Polygon","Diagonals","Jump from the one before",[85,90,95,99,104,109,114,118],[86,87,88,89],"3","Triangle","0","—",[91,92,93,94],"4","Quadrilateral","2","+2",[96,97,96,98],"5","Pentagon","+3",[100,101,102,103],"6","Hexagon","9","+4",[105,106,107,108],"7","Heptagon","14","+5",[110,111,112,113],"8","Octagon","20","+6",[102,115,116,117],"Nonagon","27","+7",[33,119,120,121],"Decagon","35","+8",{"id":123,"type":47,"variant":124,"title":125,"markdown":126},"obs-jumps","observation","The jumps go 2, 3, 4, 5, …","Each time you add one more side, the number of diagonals jumps by one more than the time before. Why? When you insert a new vertex between two old neighbours, it gets diagonals to all the other vertices except its two neighbours, and the old side between those two neighbours turns into a diagonal. Try drawing a hexagon and then pushing out one side to make a heptagon, and count the new diagonals: there are exactly **5** of them (4 from the new vertex, plus the old side that became a diagonal).",{"id":128,"type":129,"itemId":130,"prompt":131,"check":132,"hints":136,"feedback":139},"pr-dodecagon","practice","shape-and-space.investigate-dodecagon-diagonals","Use the pattern of jumps (or the vertex method) to find the number of diagonals of a **12-sided** polygon (a dodecagon).",{"kind":133,"answer":134,"tolerance":135},"number",54,0,[137,138],"Continue the table: 10 sides has 35. The next jumps are +9 and then +10.","Or: each vertex has 12 − 3 = 9 diagonals; 12 × 9 ÷ 2.",{"correct":140,"incorrect":141},"Yes: 35 + 9 = 44 for 11 sides, then 44 + 10 = 54. Check with the vertex method: 12 × 9 = 108, and 108 ÷ 2 = 54.","From each vertex 9 diagonals leave: 12 × 9 = 108 ends, so 108 ÷ 2 = 54 diagonals.",{"id":143,"type":53,"title":144,"eyebrow":145,"navLabel":146},"ch02","Can three sticks make a triangle?","Chapter 02","2 Three sticks",{"id":148,"type":43,"markdown":149},"sticks-intro","Cut straws to the lengths in the table (in cm) and try to join each set into a triangle, end to end. Before you cut, predict which sets will work.\n\nYou might think any three sticks can make a triangle if you just wiggle them enough. The experiment says otherwise.",{"id":151,"type":59,"prompt":152,"options":153,"explanation":160},"pred-2-3-6","Can you make a triangle from sticks of **2 cm, 3 cm and 6 cm**?",[154,156,158],{"id":63,"label":155},"Yes, any three sticks make a triangle",{"id":66,"label":157},"No, the two short sticks cannot reach each other",{"id":69,"label":159},"Only if you make it right-angled","**No.** Lay the 6 cm stick flat and stand the other two up from its ends. Even lying flat along the long stick, 2 + 3 = 5 cm, which does not reach across 6 cm. The two short sticks can never meet above it. The two shorter sides must add up to **more** than the longest side.",{"id":162,"type":77,"caption":163,"columns":164,"rows":169},"table-sticks","Straw triangle results",[165,166,167,168],"Stick lengths (cm)","Two shorter added","Longest","Triangle?",[170,174,177,181,185,188,191,195],[171,172,96,173],"3, 4, 5","3 + 4 = 7","Yes",[175,176,96,173],"5, 5, 5","5 + 5 = 10",[178,179,100,180],"2, 3, 6","2 + 3 = 5","No: the short sticks do not meet",[182,183,100,184],"3, 3, 6","3 + 3 = 6","No: it lies flat",[186,187,102,173],"4, 6, 9","4 + 6 = 10",[189,190,105,180],"2, 7, 4","2 + 4 = 6",[192,193,194,173],"6, 8, 13","6 + 8 = 14","13",[196,197,33,173],"1, 10, 10","1 + 10 = 11",{"id":199,"type":47,"variant":200,"title":201,"markdown":202},"aha-flat-triangle","aha","The flat case","Look at 3, 3, 6. The two short sticks add to exactly 6. They can meet, but only by lying flat along the long stick, with no space inside at all. That is not a triangle; mathematicians call it a **degenerate** triangle. So the rule is strict: the sum of the two shorter sides must be **greater than** the longest side, not equal to it.",{"id":204,"type":129,"itemId":205,"prompt":206,"check":207,"hints":219,"feedback":221},"pr-which-triangle","shape-and-space.investigate-which-triangle","Which set of lengths can make a triangle?",{"kind":208,"options":209,"correct":218},"choice",[210,212,214,216],{"id":63,"label":211},"4 cm, 4 cm, 9 cm",{"id":66,"label":213},"5 cm, 6 cm, 12 cm",{"id":69,"label":215},"7 cm, 8 cm, 14 cm",{"id":72,"label":217},"3 cm, 5 cm, 8 cm",[69],[220],"Add the two shorter lengths and compare with the longest.",{"correct":222,"incorrect":223},"Yes: 7 + 8 = 15, which is more than 14.","Check each: 4 + 4 = 8 \u003C 9 (no), 5 + 6 = 11 \u003C 12 (no), 3 + 5 = 8 = 8 (flat, no), 7 + 8 = 15 > 14 (yes).",{"id":225,"type":226,"conceptId":227,"relation":228,"explanation":229},"conn-constructing-inv","connection","constructing-angles","related_to","When you construct a triangle from three given sides with a ruler and compass, the two arcs meet only if the triangle inequality holds.",{"id":231,"type":53,"title":232,"eyebrow":233,"navLabel":234},"ch03","Adding up the angles","Chapter 03","3 Angle sums",{"id":236,"type":47,"variant":48,"title":237,"markdown":238},"tryit-tear","Tear and test","Draw three very different triangles: one tall and thin, one wide and flat, one right-angled. Cut each out, mark the three corners with dots, tear them off and fit them together, points touching, along a ruler's edge.\n\nEach time, the three corners fill a **straight angle** exactly: 180°. Now do the same with a quadrilateral's four corners. What do they make?",{"id":240,"type":59,"prompt":241,"options":242,"explanation":251},"pred-quad-sum","What do the four angles of **any** quadrilateral add up to?",[243,245,247,249],{"id":63,"label":244},"180°",{"id":66,"label":246},"270°",{"id":69,"label":248},"360°",{"id":72,"label":250},"It depends on the shape","**360°.** Draw one diagonal: it cuts any quadrilateral into **two triangles**. The angles of the two triangles together make up exactly the four corners of the quadrilateral, so the total is 2 × 180° = 360°. The four torn corners fit together into a full turn around a point.",{"id":253,"type":77,"caption":254,"columns":255,"rows":259},"table-angle-sums","Angle sums of polygons, found by cutting into triangles from one vertex",[81,80,256,257,258],"Triangles from one vertex","Angle sum","Each angle if regular",[260,263,265,268,271,274,277,280],[87,86,261,244,262],"1","60°",[92,91,93,248,264],"90°",[97,96,86,266,267],"540°","108°",[101,100,91,269,270],"720°","120°",[106,105,96,272,273],"900°","≈ 128.6°",[111,110,100,275,276],"1,080°","135°",[115,102,105,278,279],"1,260°","140°",[119,33,110,281,282],"1,440°","144°",{"id":284,"type":47,"variant":124,"title":285,"markdown":286},"obs-add-180","Each extra side adds 180°","Going down the table, every extra side adds one more triangle and so another 180°. A polygon with n sides splits into **n − 2** triangles from one vertex, so its angles add up to **(n − 2) × 180°**. You will see why this always works in Deepen.",{"id":288,"type":129,"itemId":289,"prompt":290,"check":291,"hints":294,"feedback":297},"pr-hex-angle","shape-and-space.investigate-hexagon-angle","Each angle of a regular hexagon (a honeycomb cell) is how many degrees?",{"kind":133,"answer":292,"tolerance":135,"unit":293},120,"°",[295,296],"A hexagon splits into 4 triangles from one vertex.","Share the total equally among the 6 angles.",{"correct":298,"incorrect":299},"Yes: 4 × 180° = 720°, and 720° ÷ 6 = 120°.","The angle sum is (6 − 2) × 180° = 720°. A regular hexagon has 6 equal angles: 720° ÷ 6 = 120°.",{"id":301,"type":226,"conceptId":302,"relation":303,"explanation":304},"conn-angles-inv","angles","helps_understand","The straight angle (180°) and the full turn (360°) from the Angles topic are what make the tear-and-test experiments work.",{"id":306,"type":53,"title":307,"eyebrow":308,"navLabel":309},"ch04","Always, sometimes or never?","Chapter 04","4 Always or never",{"id":311,"type":43,"markdown":312},"asn-intro","A powerful way to test your understanding of shapes is to take a statement and decide: is it **always** true, **sometimes** true, or **never** true? To show something is *sometimes* true you need one example where it works and one where it fails. To show it is *never* or *always* true you need a reason, not just a few examples.",{"id":314,"type":315,"component":316,"componentVersion":5,"config":317,"objective":390,"textAlternative":391,"help":392},"lab-asn","interactive","sort-game",{"prompt":318,"bins":319,"items":329,"seconds":135},"Is each statement always true, sometimes true or never true?",[320,323,326],{"id":321,"label":322},"always","Always true",{"id":324,"label":325},"sometimes","Sometimes true",{"id":327,"label":328},"never","Never true",[330,334,338,342,346,350,354,358,362,366,370,374,378,382,386],{"id":331,"label":332,"bin":321,"why":333},"s1","A square is a rectangle.","A square has four right angles, which is the definition of a rectangle.",{"id":335,"label":336,"bin":324,"why":337},"s2","A rectangle is a square.","Only when its sides are all equal. A door-shaped rectangle is not a square.",{"id":339,"label":340,"bin":324,"why":341},"s3","A rhombus has four right angles.","Only when it is a square. Most rhombuses have two sharp and two wide angles.",{"id":343,"label":344,"bin":327,"why":345},"s4","A triangle has two obtuse angles.","Two obtuse angles already add to more than 180°.",{"id":347,"label":348,"bin":324,"why":349},"s5","An isosceles triangle is right-angled.","The 45°, 45°, 90° triangle is; the 70°, 70°, 40° one is not.",{"id":351,"label":352,"bin":321,"why":353},"s6","An equilateral triangle is isosceles.","It has (at least) two equal sides, in fact three.",{"id":355,"label":356,"bin":324,"why":357},"s7","The diagonals of a parallelogram are equal.","Only when it is a rectangle (or square).",{"id":359,"label":360,"bin":321,"why":361},"s8","The diagonals of a rhombus cross at right angles.","This is a property of every rhombus, including squares.",{"id":363,"label":364,"bin":324,"why":365},"s9","A trapezium has two pairs of parallel sides.","Under the Indian textbook definition (at least one pair), a parallelogram counts as a trapezium, so this can happen.",{"id":367,"label":368,"bin":324,"why":369},"s10","A kite has four equal sides.","Only when the kite is a rhombus. Usually its two pairs differ.",{"id":371,"label":372,"bin":321,"why":373},"s11","A quadrilateral has angles adding to 360°.","A diagonal splits it into two triangles: 2 × 180° = 360°.",{"id":375,"label":376,"bin":324,"why":377},"s12","A polygon has more diagonals than sides.","Hexagon (9 > 6) and larger: yes. Pentagon: equal (5 = 5). Triangle and quadrilateral: fewer.",{"id":379,"label":380,"bin":321,"why":381},"s13","A pyramid has as many faces as vertices.","Both are one more than the number of base sides.",{"id":383,"label":384,"bin":327,"why":385},"s14","A prism has an odd number of vertices.","A prism has 2n vertices: always an even number.",{"id":387,"label":388,"bin":327,"why":389},"s15","A circle has exactly 4 lines of symmetry.","Every diameter is a line of symmetry, so a circle has infinitely many.","Decide whether statements about shapes are always, sometimes or never true, and justify each.","Fifteen statement cards and three bins.\n\n**Always true:** a square is a rectangle; an equilateral triangle is isosceles; the diagonals of a rhombus cross at right angles; a quadrilateral's angles add to 360°; a pyramid has as many faces as vertices.\n\n**Sometimes true:** a rectangle is a square; a rhombus has four right angles; an isosceles triangle is right-angled; a parallelogram's diagonals are equal; a trapezium has two pairs of parallel sides (under the inclusive definition); a kite has four equal sides; a polygon has more diagonals than sides.\n\n**Never true:** a triangle has two obtuse angles; a prism has an odd number of vertices (it has 2n); a circle has exactly 4 lines of symmetry (it has infinitely many).\n\nFor every *sometimes* card, find one example where it is true and one where it is false.",{"hints":393},[394,395],"For “sometimes”, look for one example that works and one that fails.","Remember the special members of each family: a square is also a rhombus, a rectangle, a parallelogram and a kite.",{"id":397,"type":47,"variant":398,"title":399,"markdown":400},"nuance-trapezium","nuance","Two definitions of trapezium","Indian textbooks and many others say a trapezium has **a pair** (at least one pair) of parallel sides; then every parallelogram is also a trapezium. Some books say **exactly one** pair; then parallelograms are left out. Neither is wrong; they are different choices. The *inclusive* choice keeps the family tree tidy, so it is used in this course. Always check which definition a question is using.",{"id":402,"type":226,"conceptId":403,"relation":303,"explanation":404},"conn-lines-inv","lines","Deciding whether sides are parallel, and whether diagonals are perpendicular, uses the ideas in the Lines topic.",{"id":406,"type":53,"title":407,"eyebrow":408,"navLabel":409},"ch05","The face-edge-vertex hunt","Chapter 05","5 F, E and V",{"id":411,"type":43,"markdown":412},"fev-hunt","Collect every box-shaped and pointed solid you can find (or use the lab). For each, count faces (F), vertices (V) and edges (E) and write them in a table. Then try combining the three numbers in different ways, adding and subtracting, to see if anything stays the same.",{"id":414,"type":315,"component":415,"componentVersion":5,"config":416,"objective":432,"textAlternative":433,"help":434},"lab-fev-all","shape-explorer",{"solids":417,"polygons":428,"modes":429},[418,419,420,421,422,423,424,425,426,427],"cube","cuboid","triangular-prism","pentagonal-prism","hexagonal-prism","triangular-pyramid","square-pyramid","cylinder","cone","sphere",[],[430,431],"count","explore","Type F, E and V for up to ten solids, then explore them to test whether F + V − E is always the same.","In **count**, a solid appears each round and you type its faces, edges and vertices. In **explore**: Drag the solid, or use the Turn and Tilt sliders, to rotate it; hidden edges show as dashed lines. You can highlight its faces, edges or vertices, and for a polyhedron the Net button unfolds it flat. The readout shows F, E, V and the Euler check.\n\nPolyhedra: cube 6 faces, 12 edges, 8 vertices; cuboid 6, 12, 8; triangular prism 5, 9, 6; pentagonal prism 7, 15, 10; hexagonal prism 8, 18, 12; triangular pyramid 4, 6, 4; square pyramid 5, 8, 5. For every one, F + V − E = **2**; for example 8 + 12 − 18 = 2.\n\nCurved solids: the lab counts the cylinder as 3 faces (2 flat and 1 curved), 2 edges, 0 vertices; the cone 2, 1, 1; the sphere 1, 0, 0. These give 1, 2 and 1, and the lab points out that Euler's formula applies to **polyhedra** only, solids with flat faces.",{"hints":435},[436,437],"Try F + V − E for the cube: 6 + 8 − 12.","Now test a pyramid and a prism. Then try the cylinder.",{"id":439,"type":77,"caption":440,"columns":441,"rows":447},"table-fev-hunt","Results of the hunt: is F + V − E always the same?",[442,443,444,445,446],"Solid","F","V","E","F + V − E",[448,450,452,454,456,459,462,465],[449,91,91,100,93],"Triangular pyramid",[451,96,96,110,93],"Square pyramid",[453,100,100,33,93],"Pentagonal pyramid",[455,96,100,102,93],"Triangular prism",[457,100,110,458,93],"Cube","12",[460,105,33,461,93],"Pentagonal prism","15",[463,110,458,464,93],"Hexagonal prism","18",[466,33,467,468,93],"Octagonal prism","16","24",{"id":470,"type":47,"variant":200,"title":471,"markdown":472},"aha-euler-found","You have found Euler’s formula","Every polyhedron in the table gives F + V − E = **2**. The Swiss mathematician Leonhard Euler wrote about it twice in 1750 and published it in 1752, so it is now called **Euler's formula**. He was not quite the first: Francesco Maurolico had written it down for the five regular solids back in 1537. It is surprising: the cube and the triangular prism look nothing alike, yet the same simple sum works for both. In Deepen you will see why it must be true for all prisms and pyramids, and meet a strange solid where it fails.",{"id":474,"type":59,"prompt":475,"options":476,"explanation":483},"pred-hept-prism","A **heptagonal prism** has 7-sided ends. Without drawing it, predict its number of edges.",[477,478,479,481],{"id":63,"label":107},{"id":66,"label":464},{"id":69,"label":480},"21",{"id":72,"label":482},"28","**21.** A prism with an n-sided base has n edges round each end and n along its length: 3n = 3 × 7 = 21. Its faces are 7 + 2 = 9 and vertices 2 × 7 = 14. Check with Euler: 9 + 14 − 21 = 2. ✓",{"id":485,"type":53,"title":486,"eyebrow":487,"navLabel":488},"ch06","Which six squares fold into a cube?","Chapter 06","6 Cube nets",{"id":490,"type":43,"markdown":491},"nets-test","Cut out 6 equal squares from card, and tape them together in different arrangements, always edge to edge. Then try to fold each arrangement into a cube. Record which work. (Squared paper makes it quicker: draw, cut round the outline, fold.)\n\nIn the lab below, each card describes an arrangement. The rows are read from top to bottom, and the description says where each square sits.",{"id":493,"type":315,"component":316,"componentVersion":5,"config":494,"objective":552,"textAlternative":553,"help":554},"lab-sort-nets",{"prompt":495,"bins":496,"items":503,"seconds":135},"Will this arrangement of 6 squares fold into a cube?",[497,500],{"id":498,"label":499},"yes","Folds into a cube",{"id":501,"label":502},"no","Does not fold",[504,508,512,516,520,524,528,532,536,540,544,548],{"id":505,"label":506,"bin":498,"why":507},"n1","Cross: a row of 4, one square above the 2nd and one below the 2nd","The row of 4 wraps into a ring; the two extra squares close the top and bottom.",{"id":509,"label":510,"bin":498,"why":511},"n2","A row of 4, one square above the 1st and one below the 4th","Any 1-4-1 net works: one extra square each side of the row of 4 always closes both ends.",{"id":513,"label":514,"bin":498,"why":515},"n3","T shape: a row of 3 on top, then 3 squares in a column below the middle one","The column plus the middle top square is a line of 4; the two side squares close the ends.",{"id":517,"label":518,"bin":501,"why":519},"n4","A row of 4 with both extra squares above it","Both extra squares fold onto the same end of the ring; the other end is left open.",{"id":521,"label":522,"bin":501,"why":523},"n5","Six squares in one straight row","Four squares make the ring; squares 5 and 6 overlap squares 1 and 2. Both ends stay open.",{"id":525,"label":526,"bin":501,"why":527},"n6","A 2 by 3 block (two rows of 3)","Any 2 by 2 block of squares folds into overlapping faces around one corner.",{"id":529,"label":530,"bin":498,"why":531},"n7","Staircase: 2 squares, then 2 shifted one step right, then 2 more shifted right","This 2-2-2 staircase is one of the 11 cube nets.",{"id":533,"label":534,"bin":498,"why":535},"n8","Two rows of 3, the lower row shifted two squares right (touching at one edge)","This 3-3 net folds: each row of 3 wraps round two sides and the pieces interlock.",{"id":537,"label":538,"bin":501,"why":539},"n9","A row of 5 with one square above the middle","Five in a row always overlaps: the 5th square lands on the 1st.",{"id":541,"label":542,"bin":498,"why":543},"n10","A row of 3 with 1 square above its left end, and a row of 2 hanging below its right end, sticking out to the right","This is a 1-3-2 net: the row of 3 and the square below its end wrap round, and the rest close the gaps.",{"id":545,"label":546,"bin":501,"why":547},"n11","A 2 by 2 block with one extra square on its left and one on its right","Any net containing a 2 by 2 block fails: those four squares cannot all be different faces of a cube.",{"id":549,"label":550,"bin":498,"why":551},"n12","A row of 4, one square above the 3rd and one below the 3rd","Another cross-like 1-4-1 net. It works for the same reason as the cross.","Predict and check which arrangements of six squares fold up into a closed cube.","Twelve cards each describe an arrangement of six equal squares joined edge to edge.\n\n**Folds into a cube:** the cross (row of 4 with squares above and below the 2nd); a row of 4 with one square above the 1st and one below the 4th; the T shape; the 2-2-2 staircase; two rows of 3 shifted so they share one edge; a row of 3 with one square above its left end and a row of 2 below its right end; a row of 4 with squares above and below the 3rd.\n\n**Does not fold:** a row of 4 with both extras on the same side; six in a row; a 2 by 3 block; a row of 5 plus one; any shape containing a 2 by 2 block.\n\nUseful tests: a row of 5 or more always fails; a 2 by 2 block always fails; in a row of 4, the two extra squares must be on opposite sides of the row.",{"simplerExplanation":555,"hints":556},"Imagine standing the middle square on the table and folding everything else up. Does each square find its own wall or the lid, with no two landing in the same place?",[557,558],"A 2 by 2 block of squares always fails.","In a row of 4, one extra square must go on each side of the row.",{"id":560,"type":47,"variant":124,"title":561,"markdown":562},"obs-opposite","Squares that end up opposite","In a working cube net, two squares that are separated by exactly one square in a straight line become **opposite faces** of the cube. That is how dice makers place the numbers: on a standard dice, opposite faces add to 7 (1 opposite 6, 2 opposite 5, 3 opposite 4). Try numbering your own net so that it folds into a proper dice.",{"id":564,"type":59,"prompt":565,"options":566,"explanation":572},"pred-how-many-nets","People have tested every possible arrangement of six squares. How many **different** cube nets are there (not counting turned or flipped copies as different)?",[567,568,569,571],{"id":63,"label":91},{"id":66,"label":100},{"id":69,"label":570},"11",{"id":72,"label":120},"**11.** There are 35 different ways to join six squares edge to edge (these shapes are called *hexominoes*), and exactly 11 of them fold into a cube. That the answer is exactly 11 is a long-established result, and you can confirm it yourself by cutting and folding. Six of the 11 have a row of four (the 1-4-1 family), three have a row of three with a pair below (1-3-2), one is the 2-2-2 staircase and one is the 3-3 shape. In Extend you will hunt for all eleven.",{"id":574,"type":53,"title":575,"eyebrow":576,"navLabel":577},"ch07","Building from views","Chapter 07","7 Views",{"id":579,"type":43,"markdown":580},"views-build","Take some identical cubes (dice, sugar cubes or cubes made from card). One person builds a small structure in secret and draws its **top**, **front** and **side** views. The other person must rebuild it from the drawings alone. It is harder than it sounds, and it is exactly the problem a builder faces when reading an architect's drawings.",{"id":582,"type":583,"title":584,"problem":585,"steps":586},"we-count-from-views","worked_example","How many cubes? Reading views","A structure of cubes has this **top view**: a row of 3 squares. Its **front view** shows columns of heights 1, 3 and 2 from left to right. How many cubes are there?",[587,588,589,590],"The top view tells us there are 3 columns standing in a single row, with nothing behind them.","Because there is only one row, the front view shows the full height of every column.","Heights: 1 + 3 + 2 = **6 cubes**.","If the top view had shown 2 rows, the front view would only show the taller column in each position, and some cubes could be hidden. Then you would need the side view too.",{"id":592,"type":315,"component":593,"componentVersion":5,"config":594,"objective":617,"textAlternative":618,"help":619},"lab-match-views","match-pairs",{"prompt":595,"mode":596,"pairs":597},"Match each set of views to the solid that makes it.","connect",[598,601,604,606,609,612,614],{"a":599,"b":600},"Top: circle. Front: rectangle","Cylinder standing on its end",{"a":602,"b":603},"Top: circle with a centre dot. Front: triangle","Cone standing on its base",{"a":605,"b":451},"Top: square. Front: triangle",{"a":607,"b":608},"Top: circle. Front: circle. Side: circle","Sphere",{"a":610,"b":611},"Top: rectangle. Front: triangle","Triangular prism lying on a rectangular face",{"a":613,"b":457},"Top: square. Front: square. Side: square",{"a":615,"b":616},"Top: rectangle. Front: circle","Cylinder lying on its side, seen end-on","Work out which solid is described by a combination of top, front and side views.","Seven view descriptions on the left, seven solids on the right. The pairs: top circle and front rectangle is a standing cylinder; top circle with a centre dot and front triangle is a cone on its base; top square and front triangle is a square pyramid; circles from every side is a sphere; top rectangle and front triangle is a triangular prism lying on a rectangular face (like a tent); squares from every side is a cube; top rectangle and front circle is a cylinder lying down with its round end facing you.\n\nNotice that the cylinder appears twice: the same solid gives different views depending on how it is placed.",{"hints":620},[621,622],"Round in one view and straight-sided in another usually means a cylinder or cone.","A triangle in the front view means the solid comes to a point or a ridge.",{"id":624,"type":47,"variant":625,"title":626,"markdown":627},"misc-views-unique","misconception","“Two views are always enough”","A square top view and a square front view could come from a cube. But they could also come from a **wedge**: a cube sliced from its top front edge down to its bottom back edge. From above, the sloping face still looks like a full square; from the front you see the full square front face. Only the **side view** (a triangle for the wedge, a square for the cube) tells them apart. That is why architects give all three views, and sometimes more.",{"id":629,"type":53,"title":630,"eyebrow":631,"navLabel":632},"ch08","Symmetry hunt","Chapter 08","8 Symmetry hunt",{"id":634,"type":47,"variant":48,"title":635,"markdown":636},"tryit-mirror","The mirror test","Stand a small mirror upright on a picture, a letter or a shape. Slide and turn it until the half you can see plus its reflection looks exactly like the whole figure. Wherever that happens, the mirror is on a **line of symmetry**. Test the letters of your own name, a leaf, a ₹10 note, the pattern on a bedsheet and a photo of a temple gopuram.",{"id":638,"type":315,"component":316,"componentVersion":5,"config":639,"objective":706,"textAlternative":707,"help":708},"lab-sort-letters",{"prompt":640,"bins":641,"items":651,"seconds":705},"How many lines of symmetry does each capital letter have (plain block capitals)?",[642,645,648],{"id":643,"label":644},"zero","No lines",{"id":646,"label":647},"one","Exactly 1 line",{"id":649,"label":650},"two","2 lines",[652,656,660,664,668,671,675,679,682,686,690,694,698,702],{"id":653,"label":654,"bin":646,"why":655},"la","A","A vertical line through the top point.",{"id":657,"label":658,"bin":646,"why":659},"lb","B","A horizontal line through the middle.",{"id":661,"label":662,"bin":649,"why":663},"lh","H","Vertical and horizontal lines through the centre.",{"id":665,"label":666,"bin":649,"why":667},"lx","X","Vertical and horizontal (in a font where the arms are at equal angles).",{"id":669,"label":443,"bin":643,"why":670},"lf","F has arms only on the right, and only at the top and middle.",{"id":672,"label":673,"bin":643,"why":674},"ls","S","No mirror line, although S looks the same after a half turn.",{"id":676,"label":677,"bin":646,"why":678},"lm","M","A vertical line down the middle.",{"id":680,"label":445,"bin":646,"why":681},"le","A horizontal line through the middle arm.",{"id":683,"label":684,"bin":649,"why":685},"li","I","A plain I with top and bottom bars has a vertical and a horizontal line.",{"id":687,"label":688,"bin":643,"why":689},"lz","Z","No mirror line; like S and N it has half-turn symmetry.",{"id":691,"label":692,"bin":646,"why":693},"lt","T","A vertical line down the stem.",{"id":695,"label":696,"bin":643,"why":697},"lr","R","The leg makes the bottom differ from the top and the left from the right.",{"id":699,"label":700,"bin":649,"why":701},"lo","O","An oval O has exactly 2 lines. (A perfectly round O would have infinitely many!)",{"id":703,"label":704,"bin":646,"why":659},"lc","C",60,"Find how many mirror lines each capital letter has, against the clock.","Fourteen capital letters, three bins, and a 60-second timer.\n\n**No lines:** F, S, Z, R. **Exactly one line:** A, M, T (vertical); B, E, C (horizontal). **Two lines:** H, I, X and O (both vertical and horizontal).\n\nS and Z have no mirror line, but they look the same when turned upside down: that is rotational symmetry, not line symmetry. The answers assume plain block capitals; fancy fonts can change them. An oval O has 2 lines; a perfectly round O, a circle, would have infinitely many.",{"hints":709},[710,711],"Test a vertical mirror first, then a horizontal one.","S, N and Z are the tricky ones: turning is not the same as reflecting.",{"id":713,"type":59,"prompt":714,"options":715,"explanation":720},"pred-parallelogram-sym","Cut out a parallelogram that is not a rectangle or rhombus (for example with sides 8 cm and 5 cm and angles 60° and 120°). How many lines of symmetry will you find by folding?",[716,717,718,719],{"id":63,"label":88},{"id":66,"label":261},{"id":69,"label":93},{"id":72,"label":91},"**0.** Try folding it along the diagonals and through the middles of the sides: the halves are the same size but never land on each other. Yet a parallelogram does look the same after a **half turn** around its centre. That is rotational symmetry. Having no mirror line does not mean a shape has no symmetry at all.",{"id":722,"type":53,"title":723,"eyebrow":724,"navLabel":725},"ch09","Measuring round things: finding π","Chapter 09","9 Finding π",{"id":727,"type":43,"markdown":728},"pi-intro","Wrap a thread once around a bangle, mark it, straighten it along a ruler: that is the **circumference**. Measure straight across the middle: that is the **diameter**. Now divide circumference by diameter. Do this for several round objects and compare.",{"id":730,"type":77,"caption":731,"columns":732,"rows":737},"table-pi","A class’s measurements (in cm) and the ratio circumference ÷ diameter",[733,734,735,736],"Object","Diameter","Circumference","C ÷ d",[738,743,748,752,757,760],[739,740,741,742],"Glass bangle","6.5","20.5","3.15",[744,745,746,747],"Steel plate","26","81.5","3.13",[749,750,751,747],"Bucket rim","30","94",[753,754,755,756],"Cycle wheel","70","220","3.14",[758,105,759,756],"Tumbler rim","22",[761,750,762,742],"Clock face","94.5",{"id":764,"type":47,"variant":200,"title":765,"markdown":766},"aha-pi","Always a little more than 3","Big wheel or small bangle, the ratio comes out close to **3.14** every time. The small differences are measuring errors (threads stretch, rulers are read to the nearest millimetre). The true value is a never-ending decimal, 3.14159…, called **π** (pi). Every circle in the universe has the same ratio: circumference ÷ diameter = π.",{"id":768,"type":59,"prompt":769,"options":770,"explanation":779},"pred-double-d","If you double the diameter of a circle, what happens to its circumference?",[771,773,775,777],{"id":63,"label":772},"It stays the same",{"id":66,"label":774},"It doubles",{"id":69,"label":776},"It becomes 4 times as long",{"id":72,"label":778},"It increases by 3.14 cm","**It doubles.** Circumference = π × diameter, so twice the diameter means twice the circumference. A 140 cm wheel travels twice as far per turn as a 70 cm one. (What becomes 4 times as big is the **area** inside, but that is a story for another topic.)",{"id":781,"type":47,"variant":782,"title":783,"markdown":784},"model-limit-thread","model_limit","What the thread method leaves out","Measuring with a thread can only give π to about two decimal places: 3.1 or 3.14 on a good day. To get more digits, mathematicians stopped measuring and started **calculating**. Around 499 CE, the Indian mathematician **Aryabhata** gave a value equivalent to 3.1416, correct to four decimal places. Today computers have calculated trillions of digits.",{"id":786,"type":53,"title":787,"eyebrow":788,"navLabel":789},"ch10","Which shapes tile a floor?","Chapter 10","10 Tiling",{"id":791,"type":43,"markdown":792},"tile-intro","Look at floor tiles in homes, railway stations and temples. They fit together with **no gaps and no overlaps**. A pattern like this is called a **tiling** or **tessellation**. Which regular polygons can tile a floor all by themselves?\n\nAt every point where tiles meet, the angles must add up to exactly **360°**, a full turn. So a regular polygon can tile on its own only if its angle divides 360° exactly.",{"id":794,"type":77,"caption":795,"columns":796,"rows":801},"table-tiling","Can copies of one regular polygon tile a floor?",[797,798,799,800],"Regular polygon","Each angle","360° ÷ angle","Tiles alone?",[802,803,804,807,808],[87,262,100,173],[92,264,91,173],[97,267,805,806],"≈ 3.33","No, gaps are left",[101,270,86,173],[111,276,809,806],"≈ 2.67",{"id":811,"type":59,"prompt":812,"options":813,"explanation":822},"pred-octagon-tile","Regular octagons cannot tile a floor alone. But many Indian floors have octagon tiles. What fills the gaps?",[814,816,818,820],{"id":63,"label":815},"Small triangles",{"id":66,"label":817},"Small squares",{"id":69,"label":819},"Small pentagons",{"id":72,"label":821},"Nothing: the gaps are left empty","**Small squares.** Two octagon angles (135° + 135° = 270°) leave 90° at each meeting point: exactly the angle of a square. So octagons and squares together tile the floor perfectly. Look for this pattern in old houses, station platforms and hotel lobbies.",{"id":824,"type":315,"component":415,"componentVersion":5,"config":825,"objective":838,"textAlternative":839,"help":840},"lab-explore-tiles",{"solids":826,"polygons":827,"modes":836},[],[828,829,830,831,832,833,834,835],"triangle","square","pentagon","hexagon","octagon","rhombus","parallelogram","trapezium",[837],"sort","Answer yes-or-no questions about the sides and angles of polygons that appear in floor tilings, and name each shape.","An 8-round game with eight tile shapes: a triangle (drawn scalene), a square, a regular pentagon, a regular hexagon, a regular octagon, a rhombus, a parallelogram and a trapezium.\n\nEach round shows one shape and asks you to **name** it or asks a yes-or-no question: *Is it a quadrilateral? Does it have a pair of parallel sides? Are all its sides equal? Does it have a right angle?*\n\nUseful facts: the square, rhombus, parallelogram and trapezium are quadrilaterals, and all four have parallel sides. All sides are equal in the square, the rhombus and the regular pentagon, hexagon and octagon. Only the square has a right angle.\n\nThe link to tiling: every triangle and every quadrilateral tiles a floor (their angles, 180° or 360°, can be fitted round a point), while among regular polygons only the triangle, square and hexagon tile alone. The pentagon (108°) and octagon (135°) leave gaps.",{"hints":841},[842,843],"Parallel sides never meet, however far you extend them.","A rhombus has equal sides but no right angle (unless it is a square).",{"id":845,"type":47,"variant":846,"title":847,"markdown":848},"example-honeycomb-inv","example","Why hexagons win for bees","Of the three regular polygons that tile (triangle, square, hexagon), the hexagon is the most **round**. For the same area, a regular hexagon has a smaller perimeter than a square or triangle, so bees need less wax. For cells of area 1 cm², the perimeter is about 4.56 cm for a triangle, 4.00 cm for a square and 3.72 cm for a hexagon.",{"id":850,"type":43,"markdown":851},"perim-same","A related puzzle: with 24 m of fencing, which rectangle encloses the most ground? Try every whole-number rectangle with perimeter 24 m:",{"id":853,"type":77,"caption":854,"columns":855,"rows":860},"table-perim-24","Rectangles with a perimeter of 24 m",[856,857,858,859],"Length (m)","Breadth (m)","Perimeter (m)","Area (m²)",[861,862,863,864,866,867],[261,570,468,570],[93,33,468,112],[86,102,468,116],[91,110,468,865],"32",[96,105,468,120],[100,100,468,868],"36",{"id":870,"type":47,"variant":124,"title":871,"markdown":872},"obs-square-best","The squarest shape wins","The areas grow from 11 m² to 36 m² as the rectangle becomes more square. Among rectangles with the same perimeter, the **square** encloses the most. Go further and allow any shape, and the **circle** beats everything: the same idea as the bees' hexagons, and why so many water tanks and grain silos are round.",{"id":874,"type":875,"title":876,"terms":877},"glossary-investigate","glossary","Words from the investigations",[878,881,885,888,892,894,897,900,903,906],{"term":879,"meaning":880},"Conjecture","A pattern or rule that seems true from examples but has not been proved yet.",{"term":882,"meaning":883,"example":884},"Counterexample","One example that shows a statement is false.","A 6 cm by 4 cm rectangle is a counterexample to “every rectangle is a square”.",{"term":886,"meaning":887},"Triangle inequality","In any triangle, the two shorter sides together are longer than the longest side.",{"term":889,"meaning":890,"example":891},"Degenerate triangle","Three points in a straight line: a “triangle” that has collapsed flat.","Sticks 3, 3 and 6 cm.",{"term":257,"meaning":893},"The total of all the interior angles of a polygon: (n − 2) × 180°.",{"term":895,"meaning":896},"Euler’s formula","For any (simple) polyhedron, faces + vertices − edges = 2.",{"term":898,"meaning":899},"Hexomino","A shape made of six equal squares joined edge to edge. There are 35 of them; 11 fold into a cube.",{"term":901,"meaning":902},"Pi (π)","The ratio circumference ÷ diameter, the same for every circle: about 3.14159.",{"term":904,"meaning":905},"Tessellation (tiling)","A pattern of shapes covering a surface with no gaps and no overlaps.",{"term":907,"meaning":908},"Rotational symmetry","A shape has it if it looks the same after a turn of less than a full turn about its centre.",{"id":910,"type":911,"title":912,"questions":913},"quiz-investigate","quiz","What did the investigations show?",[914,923,936,946,959,968,981,994,1007,1016],{"itemId":915,"prompt":916,"options":917,"correct":69,"why":922},"shape-and-space.investigate-q-decagon","How many diagonals does a decagon have?",[918,919,920,921],{"id":63,"label":112},{"id":66,"label":116},{"id":69,"label":120},{"id":72,"label":754},"10 × 7 ÷ 2 = 35. (70 counts every diagonal twice.)",{"itemId":924,"prompt":925,"options":926,"correct":69,"why":935},"shape-and-space.investigate-q-sticks","Sticks of 4 cm and 9 cm are two sides of a triangle. Which could be the third side?",[927,929,931,933],{"id":63,"label":928},"4 cm",{"id":66,"label":930},"5 cm",{"id":69,"label":932},"8 cm",{"id":72,"label":934},"13 cm","The third side must be more than 9 − 4 = 5 and less than 9 + 4 = 13. Only 8 cm fits.",{"itemId":937,"prompt":938,"options":939,"correct":69,"why":945},"shape-and-space.investigate-q-pent-sum","What do the angles of a pentagon add up to?",[940,941,943,944],{"id":63,"label":248},{"id":66,"label":942},"480°",{"id":69,"label":266},{"id":72,"label":269},"A pentagon splits into 3 triangles from one vertex: 3 × 180° = 540°.",{"itemId":947,"prompt":948,"options":949,"correct":66,"why":958},"shape-and-space.investigate-q-asn","Which statement is **never** true?",[950,952,954,956],{"id":63,"label":951},"A rhombus is a square",{"id":66,"label":953},"A triangle has two right angles",{"id":69,"label":955},"A kite has four equal sides",{"id":72,"label":957},"A rectangle is a square","Two right angles already make 180°, leaving nothing for the third angle. The other three are sometimes true.",{"itemId":960,"prompt":961,"options":962,"correct":66,"why":967},"shape-and-space.investigate-q-euler","A polyhedron has 10 faces and 16 vertices. How many edges does it have?",[963,964,965,966],{"id":63,"label":759},{"id":66,"label":468},{"id":69,"label":745},{"id":72,"label":482},"F + V − E = 2, so 10 + 16 − E = 2 and E = 24. (This is an octagonal prism.)",{"itemId":969,"prompt":970,"options":971,"correct":72,"why":980},"shape-and-space.investigate-q-net","Why does a 2 by 3 block of six squares fail to fold into a cube?",[972,974,976,978],{"id":63,"label":973},"It has too few squares",{"id":66,"label":975},"The squares are the wrong size",{"id":69,"label":977},"It does fold into a cube",{"id":72,"label":979},"It contains a 2 by 2 block, whose squares overlap when folded","Four squares around one point fold so that two of them land on the same face. Any arrangement with a 2 by 2 block fails.",{"itemId":982,"prompt":983,"options":984,"correct":66,"why":993},"shape-and-space.investigate-q-pi","A plate has a diameter of 20 cm. About how long is its rim?",[985,987,989,991],{"id":63,"label":986},"40 cm",{"id":66,"label":988},"63 cm",{"id":69,"label":990},"100 cm",{"id":72,"label":992},"31 cm","C ≈ 3.14 × 20 = 62.8 cm, about 63 cm.",{"itemId":995,"prompt":996,"options":997,"correct":63,"why":1006},"shape-and-space.investigate-q-tile","Why can regular pentagons not tile a floor by themselves?",[998,1000,1002,1004],{"id":63,"label":999},"Their 108° angle does not divide 360° exactly",{"id":66,"label":1001},"They have too many sides",{"id":69,"label":1003},"They are not regular",{"id":72,"label":1005},"They can","360 ÷ 108 ≈ 3.33, not a whole number, so copies cannot fill the space round a point.",{"itemId":1008,"prompt":1009,"options":1010,"correct":69,"why":1015},"shape-and-space.investigate-q-views","The top view of a stack of cubes is 4 squares in a 2 by 2 block. What is the smallest possible number of cubes?",[1011,1012,1013,1014],{"id":63,"label":261},{"id":66,"label":93},{"id":69,"label":91},{"id":72,"label":110},"Each square in the top view is a column with at least one cube, so at least 4.",{"itemId":1017,"prompt":1018,"options":1019,"correct":69,"why":1028},"shape-and-space.investigate-q-rect","Which rectangle with a perimeter of 20 m has the largest area?",[1020,1022,1024,1026],{"id":63,"label":1021},"9 m by 1 m",{"id":66,"label":1023},"7 m by 3 m",{"id":69,"label":1025},"5 m by 5 m",{"id":72,"label":1027},"6 m by 4 m","Areas: 9, 21, 24, 25 m². The square wins.",{"id":1030,"type":1031,"prompt":1032},"reflect-inv","reflection","Which investigation surprised you most? Write down what you predicted, what actually happened, and one new question the result made you want to test.",{"id":1034,"type":1035,"title":1036,"points":1037},"cheat-investigate","summary","Cheat sheet",[1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048],"Test before you trust: predict, try examples, look for a pattern, then hunt for a **counterexample**.","Diagonals: 0, 2, 5, 9, 14, 20, 27, 35 for 3 to 10 sides; the jumps grow 2, 3, 4, 5, …","**Triangle inequality:** the two shorter sides must add to **more** than the longest. Equal gives a flat (degenerate) triangle.","Angle sums: triangle 180°, quadrilateral 360°, pentagon 540°, hexagon 720°. Each extra side adds 180°: **(n − 2) × 180°**.","*Always, sometimes, never:* a square is always a rectangle; a rectangle is sometimes a square; a triangle never has two obtuse angles.","For every polyhedron tested, **F + V − E = 2** (Euler’s formula). It does not apply neatly to curved solids.","Exactly **11** of the 35 hexominoes fold into a cube. A row of 5 or a 2 by 2 block always fails.","Views: the top view shows columns; front and side views show heights. You often need all three.","Circumference ÷ diameter ≈ **3.14** (π) for every circle. Double the diameter, double the circumference.","Only equilateral triangles, squares and regular hexagons tile a floor alone (their angles divide 360°). Octagons need squares.","For a fixed perimeter, the squarest rectangle has the biggest area; a circle beats every shape.",{"id":1050,"type":226,"conceptId":1051,"relation":228,"explanation":1052},"conn-patterns-inv","patterns","The diagonal counts 0, 2, 5, 9, 14, … and the angle sums 180°, 360°, 540°, … are number patterns hiding inside shapes.",{"id":1054,"type":226,"conceptId":1055,"relation":228,"explanation":1056},"conn-data-inv","data-handling","Measuring many circles and averaging C ÷ d is a data investigation: the mean smooths out measuring errors.",{"id":1058,"type":1059,"sourceIds":1060},"sources-investigate","sources",[1061,1062,1063,1064,1065,1066,1067,1068,1069],"shape-and-space-ncert-class6","shape-and-space-ncert-class7","shape-and-space-ncert-class8","shape-and-space-mathsisfun-euler","shape-and-space-mathsisfun-quadrilaterals","shape-and-space-mathsisfun-circle","shape-and-space-wiki-net","shape-and-space-wiki-honeycomb","shape-and-space-eppstein-euler",[1061,1062,1063,1064,1065,1066,1067,1068,1069],"needs_review",{"generatedBy":1073,"notes":1074},"claude-code","Draft generated locally; pending owner review.","c29fb405f3f670ff0f869eacf46a459eb40769ddb574b017ad0b09db8c44c166",{"logic:practice":1077,"component:sort-game@1":1078,"component:shape-explorer@1":1079,"component:match-pairs@1":1080,"source:shape-and-space-eppstein-euler":1081,"source:shape-and-space-mathsisfun-circle":1082,"source:shape-and-space-mathsisfun-euler":1083,"source:shape-and-space-mathsisfun-quadrilaterals":1084,"source:shape-and-space-ncert-class6":1085,"source:shape-and-space-ncert-class7":1086,"source:shape-and-space-ncert-class8":1087,"source:shape-and-space-wiki-honeycomb":1088,"source:shape-and-space-wiki-net":1089},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","a8965f19a8548e866e5fcd7f4fec4a9adac35ad54d43c9d5c416cdf3348e6198","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","95e13e2d031c58f703234cbb201c5fb309036fe3e69ac294cefe54ce2397c51f","ff337c822df6bd0ba5ef54450ce49945d4ef414438058441e656ee73ca685947","512952430fd4d2cad9d5ad14eb51fe077bf3637c74e2c604e6458fe725b018b6","292166035f591d3d313675b23dcd587f26add442ae362358b916e3e15caa24b4","e1821bd507663f06793be51247da07646b65ba5a734a31c173a63c03f426f078","20d54bcdc9a5cc5930102f965a7435de619c7bc9731d1de2ff67ae1d0307ca5c","9516472286b1156fa1275e7ea5b6574beff3acc596b8dfe8563f5c446cb7d6b6","4336a097e06982ba6b142ea63fb6f280c2335bc502b4e3e9f754af2d63e6771f","d6c8b2e7ef98f5aa4726bed69ebb23d330aeb205ee01d6f8d2b426062a952f6e",{"state":1091,"reviewer":1092,"selfReview":1093,"reviewedAt":1094,"method":1095},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597657]