Number pyramids: what the top cell is made of

What to assume they know

• Adding and subtracting whole numbers, and reading subtraction as the reverse of addition
• Writing an unknown as a letter and building an expression from it
• Turning a stated relationship into an equation and solving a one-unknown equation
• Collecting like parts of an expression, so that a letter met twice becomes twice that letter
• The Virahāṅka-Fibonacci sequence as introduced earlier: each term is the total of the two before it (this chapter restates it starting 1, 2, 3, 5)
• Substituting numbers into an expression to check it

What they should be able to do

• State the pyramid rule and use it to fill a pyramid upward from a complete bottom row
• Fill a pyramid downward from partial information, using subtraction, and say when that is enough
• Recognise a pyramid that subtraction alone cannot unlock, and explain why
• Write letters into the blank cells of a pyramid, generate the equations the rule forces, and solve them
• Derive the top-cell expression for a pyramid of two, three and four rows in terms of the bottom row
• Explain the weights 1, 2, 1 and 1, 3, 3, 1 by counting upward routes from each bottom cell to the apex
• Compute a top cell directly from a bottom row without building the middle rows
• Predict what happens when the bottom row holds the first few Virahāṅka-Fibonacci numbers, and justify it from the sequence's defining property
• State which term of that sequence sits at the apex of an n-row pyramid, and why

Where it usually goes wrong

• "The apex is the bottom row added up." It is not, and the chapter's opening figure refutes it immediately: 1, 9 and 4 total 14 and the apex reads 23. The gap is the middle cell, counted a second time.
• "So the weights go 1, 2, 3, 4, …" They do not. For four cells they are 1, 3, 3, 1. A student who has only seen the three-row case will guess an arithmetic run; route counting is what stops the guessing.
• "You can always fill a pyramid downward by subtracting." Only when a known cell sits directly next to or above another. The pyramid with apex 60 has three known cells and no usable pair, which is why the chapter reaches for letters at that exact moment.
• "One letter cannot possibly be enough." It is, because the rule itself manufactures the other equations. This is the moment the chapter is really teaching, and it is easy to skate past.
• "The two middle equations are extra information." They are the same rule applied twice, and adding them is what eliminates two unknowns at once. Students often solve the three equations separately and lose the shortcut.
• "Fibonacci going in and Fibonacci coming out is a coincidence." It follows from the definition. Every cell above the bottom row is two neighbours added, and two neighbours in this sequence always add to the next term.
• "A 29-row pyramid has the 29th term on top." It has the 57th. The rows shift two places along the sequence each time you go up one, not one place.
• "A pyramid with a blank in the bottom row is impossible." Three of the four four-row pyramids in this section have exactly that, and all of them close.

Questions to check understanding

• Fill a pyramid upward from a complete bottom row
• Fill a pyramid downward from an apex and one edge, and state the order the cells had to be found in
• Given a pyramid whose known cells do not touch, set up the equations and solve
• Write the top-cell expression for a pyramid of a stated number of rows
• Compute an apex from a bottom row without building the intermediate rows, and state the weights used
• Given an apex and all but one bottom cell, find the missing cell
• Justify, rather than illustrate, that a pyramid built on consecutive Virahāṅka-Fibonacci numbers contains only terms of that sequence
• Name the sequence position of the apex for a pyramid of a stated depth

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. No answer to any exercise the chapter sets the reader is printed anywhere in Part II pp.135–147, and Part II has no answer-key appendix. (The chapter does print worked answers to its own demonstrations — the 291 decode resolved to 25th December on Part II p.137, the apex-10 and apex-60 pyramids shown completed on Part II pp.138–139, the sum-36 block given as 5, 6, 12, 13 on Part II p.141, and the worked algebra grid resolving to 9 and 5 on Part II p.142 — so do not say the chapter prints no answers at all.)

• The introductory pyramid (Part II §6.3, p.137, three rows). Bottom row 1, 9, 4; middle row 10, 13; apex 23. Fully printed, and worth pausing on because the bottom row adds to 14 while the apex reads 23.
• Three pyramids set for the reader, tops blank (Part II §6.3, p.138). Their bottom rows are 6 and 2; then 3, 4 and 3; then 5, 4, 5 and 0 — two, three and four cells respectively, so the three of them walk up the row sizes on purpose. Verified apexes: 8; 14; 32. The last one is the one to dwell on, because its bottom row adds to 14 and its apex is 32.
• The pyramid worked downward, in four printed stages (Part II §6.3, p.138). Three rows. Given: apex 10, the left cell of the middle row 4, the left cell of the bottom row 1. The page fills it by subtraction in this order — the middle row's right cell as 10 − 4 = 6, then the bottom middle as 4 − 1 = 3, then the bottom right as 6 − 3 = 3. Completed pyramid: bottom 1, 3, 3; middle 4, 6; apex 10. In the artwork each newly found cell is set in a different colour from the given ones; that colour coding is the figure's whole pedagogy and should survive into the figure.
• The pyramid subtraction cannot start on (Part II §6.3, p.138, three rows). Given: apex 60, bottom left 12, bottom right 8, with the whole middle row and the bottom centre blank. The page asks where one would even begin. Nothing here sits directly above or beside a known partner, which is exactly the point.
• The same pyramid solved with letters (Part II §6.3, p.139). The page names the middle row's two cells and the bottom centre, and prints the three equations the rule forces: the two middle cells total 60; the left middle cell is 12 plus the bottom centre; the right middle cell is the bottom centre plus 8. Adding the last two gives 20 plus twice the bottom centre equal to 60, so the bottom centre is 20, and the completed pyramid is printed: bottom 12, 20, 8; middle 32, 28; apex 60.
• Three four-row pyramids set for the reader (Part II §6.3, p.139). All three are four rows deep. The given cells, read off the printed page and confirmed on the printed page:
  • First: apex 50; the right cell of the second row 22; bottom row 4, blank, 6, blank. Verified: bottom 4, 9, 6, 1; third row 13, 15, 7; second row 28, 22.
  • Second: apex blank; the left cell of the second row 40; the right cell of the third row 9; bottom row 5, blank, 7, blank. Verified: bottom 5, 14, 7, 2; third row 19, 21, 9; second row 40, 30; apex 70.
  • Third: apex 35; the right cell of the third row 7; bottom row 3, 5, blank, blank. Verified: bottom 3, 5, 5, 2; third row 8, 10, 7; second row 18, 17. These three are not graded, and they should not be presented as though they were: none of the three can be closed by subtraction alone. Each reduces to a single unknown that appears twice in a known total, so each needs exactly the move the chapter has just taught on the apex-60 pyramid — name a bottom cell, add the two forced expressions, halve. First: 50 − 22 = 28, then 28 = (4 + x) + (x + 6) = 2x + 10, so x = 9. Second: 9 − 7 = 2, then 40 = (5 + x) + (x + 7) = 2x + 12, so x = 14. Third: 3 + 5 = 8, then x + y = 7 with 3x + y = 17, so 2x = 10 and x = 5. That they are all the same difficulty is a better point than a grading would be, because it is why §6.3 sets them right here: three consecutive rehearsals of the one method.
• The two-row pyramid in letters (Part II §6.3, p.139) and the three-row pyramid in letters (Part II §6.3, p.140). Both printed in full. The three-row figure shows the middle row as the two adjacent pairs added, and the apex as the outer two cells plus twice the middle one.
• Figure it Out item 1 (Part II §6.3, p.140). Three three-cell bottom rows: 4, 13, 8; then 7, 11, 3; then 10, 14, 25. The item forbids building the pyramid. Verified apexes: 38; 32; 63.
• Figure it Out item 2 (Part II §6.3, p.140) asks for the four-row top-cell expression. Verified: the two outer bottom cells once each and the two inner ones three times each.
• Figure it Out item 3 (Part II §6.3, p.140). Three four-cell bottom rows: 8, 19, 21, 13; then 7, 18, 19, 6; then 9, 7, 5, 11. Verified apexes: 141; 124; 56.
• Figure it Out items 4, 5 and 6 (Part II §6.3, p.140), all on the Virahāṅka-Fibonacci sequence, which the item block restates as 1, 2, 3, 5 with each term the total of the two before it. Verified: a three-row pyramid on 1, 2, 3 gives middle row 3, 5 and apex 8 — every cell is in the sequence. A four-row pyramid on 1, 2, 3, 5 gives 3, 5, 8, then 8, 13, then 21. For a 29-row pyramid the apex is the 57th term of the sequence, and in general an n-row pyramid puts the term numbered 2n − 1 at the apex. The reason is the sequence's own rule: adding two neighbours in the sequence lands you on the next term, so each row of the pyramid is again a stretch of the sequence, shifted two places along.
• The chapter's own hinge question (Part II §6.3, p.139): what connects the bottom row to the apex. Everything from section 7 onward exists to answer it.

Figures to have open

• Blank pyramid frames of two, three and four rows, fillable cell by cell. The chapter's own figures (Part II pp.137–140) are exactly these; redraw them rather than reproducing the printed art.
• The route-counting diagram for section 9. This is not in the chapter and is the explanation's main contribution; it must show every distinct upward path, not just the totals, or the argument becomes another assertion.
• The three four-row pyramids from Part II p.139 with their given cells only, so the student can attempt them. Standard schematic.
• A strip of the Virahāṅka-Fibonacci sequence with positions numbered, against which the cells of a filling pyramid can be highlighted. Standard schematic; the numbering is what makes section 12 checkable.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 6, "Algebra Play", §6.3 "Number Pyramids", Part II pp.137–140. The section opens at the foot of Part II p.137 and its Figure it Out block occupies the lower two-thirds of Part II p.140.
• The four-stage downward fill is a single composite figure spanning most of Part II p.138; the letters-in-the-blanks derivation occupies the top half of Part II p.139.
• The Virahāṅka-Fibonacci sequence is restated inside the Figure it Out block on Part II p.140, between items 3 and 4, rather than in the running text.
• The chapter's SUMMARY (Part II p.147) lists number pyramids among the things algebra was applied to here.
• Backward pointer: the sequence itself is not developed in this chapter, which asks the reader to recall it.