Finding an unknown by reasoning about what must balance

What to assume they know

• An equation is a claim of equality, not an instruction to compute — a level bar asserts that two sides agree; the ringed number is the whole weight below it
• Adding and subtracting whole numbers, and dividing a total into equal parts
• Reading a quantity off a picture and holding several at once
• Working with kilograms as a unit
• Comparing two expressions by reasoning, not by evaluating — comparing two expressions by reasoning about their parts instead of evaluating both

What they should be able to do

• Work out an unknown weight from a hanging figure and say which step did the work
• Use the fact that equal removals from both sides leave the agreement intact
• Handle a figure in which the unknown appears on both sides by removing a matched copy from each
• Chain results: use an unknown found early in a figure to unlock a later one
• Read a nested hanger, where one branch is itself a bar with two things on it
• Explain why the same reasoning works whether the numbers are 5 and 7 or 90 and 500
• Distinguish reasoning that forces an answer from guessing that happens to land on one
• State what this picture-based method still cannot do, and why symbols are needed next

Where it usually goes wrong

• "You find the unknown by trying numbers until one fits." Nothing in §7.1 works that way, and the chapter says so out loud at the top of §7.2 when it contrasts these figures with trial and error (Part II, §7.2, p.169). Each figure forces its answer.
• "A level bar means each side weighs half the ring number." True of the hangers in Figs. 7.1–7.5, where the bar is drawn level and the ring gives the total — and it is worth stating in that form. But it is not what "balanced" means in general: the opening picture on p.164 hangs 4 and 3 under a ring marked 7, and it tips. The halving is a consequence of the bar being level, not the definition of the device.
• "Fig. 7.10 needs a different idea because the sacks are on both pans." It needs the same idea used once more. The book's hint says as much.
• "Big numbers need a different method." Fig. 7.12 is 90 against 60 with a 500 kg block, and the move is identical to Fig. 7.9. If the explanation makes it look harder, the point of printing it is lost.
• "An answer is right because it comes out to a whole number." Whole answers here are a design choice by the authors, not evidence. The instruction on p.165 asks for a reason, not for a tidy value.
• "Once you can do the pictures you can do the algebra." Not yet. The pictures hand you a physical excuse for each move; §7.2 has to argue the same moves from the meaning of the equals sign, without a scale to lean on.

Questions to check understanding

• Given a loaded balance, find the unknown and state which removal made it possible
• Given a balance with the unknown on both pans, reduce it to the unknown on one pan
• Explain in words why removing the same amount from both sides is legitimate
• Produce two different routes to the same unknown and compare them
• Given a hanger with two unknowns, say which one must be found first and why
• Rewrite a loaded balance as an equation using a letter for the unknown — the chapter asks exactly this for Figs. 7.6, 7.7, 7.8, 7.9, 7.10 and 7.11 (Part II, §7.1, p.168)
• Frame and solve five equations of your own (Part II, §7.1, p.168, Math Talk)

Examples worth working on the board

Part II prints no answers to any of these, so every number below is either printed on the page or derived here from the checked artwork, and each is marked.

• The device itself (Part II, §7.1, p.164). Checked against the printed page. Two pictures establish the rules before any question is asked: a level bar under a ring marked 4 carrying 2 and 2, and a tilted bar under a ring marked 7 carrying 4 on the low side and 3 on the high side. Printed inputs. Every later hanger in §7.1 is drawn level.
• Fig. 7.1 (Part II, §7.1, p.164). Checked against the printed page. Printed inputs: ring 16; leaf = 3; left string carries leaf, bud, leaf; right string carries a single flower; blanks for bud and flower. Derived here: each side carries 8, so the flower is 8 and the bud is 2.
• Fig. 7.2 (Part II, §7.1, p.164). Checked against the printed page. Printed inputs: ring 24; starfish = 2; left string carries starfish, striped fish, starfish; right string carries a striped fish and a grey fish; blanks for the two fish. Derived here: each side carries 12, so the striped fish is 8 and the grey fish is 4. This is the first figure that has to be worked in two stages.
• Fig. 7.3 (Part II, §7.1, p.164). Checked against the printed page. Printed inputs: ring 8; a book hangs on the left; on the right a second, shorter bar hangs from the first and carries two identical currency notes; blanks for book and note. Derived here: the book is 4 and each note is 2. The nested bar is the point — a branch can itself be a two-sided claim.
• Fig. 7.4 (Part II, §7.1, p.165). Checked against the printed page. Printed inputs: ring 18; sun = 5; left string carries a sun and four clouds; right string carries three lightning bolts; blanks for cloud and bolt. Derived here: cloud 1, bolt 3. Worth showing because the small numbers make the two-stage reasoning easy to narrate.
• Fig. 7.5 (Part II, §7.1, p.165). Checked against the printed page. Printed inputs: ring 40; left string carries a crown and five gems; right string carries four crowns; blanks for crown and gem. Derived here: crown 5, gem 3. Here the right side has to be cracked first — a useful reversal.
• Figs. 7.6, 7.7, 7.8 (Part II, §7.1, p.165). Checked against the printed page. Three hand-held level bars with no ring and no total. Fig. 7.6: three bread slices against two fried eggs, bread = 2. Fig. 7.7: four stars against a ring-shaped item, a star and a second ring-shaped item, star = 4. Fig. 7.8: a watermelon slice against a banana, an orange and a second banana, watermelon = 10 and orange = 4. All printed inputs. The chapter itself later writes 2e = 6 for the first and 4 + 2y = 16 for the second (Part II, §7.1, p.168).
• Fig. 7.9 (Part II, §7.1, p.165). Checked against the printed page. A two-pan balance. Left pan: one sack and a 2 kg weight. Right pan: a 10 kg and a 2 kg weight. Printed inputs. The matched 2 kg on each pan is the whole lesson.
• Fig. 7.10 (Part II, §7.1, p.165). Checked against the printed page. Left pan: two sacks. Right pan: a 10 kg weight, a 4 kg weight and one sack. The text states that all sacks weigh the same, and the printed hint tells the reader to take one sack off each pan. Printed inputs.
• Fig. 7.11 (Part II, §7.1, p.166). Checked against the printed page. Left pan: five sacks. Right pan: two 10 kg weights, a 1 kg weight and two sacks. The printed hint asks whether objects can be removed so that the sacks end up on one pan only. Printed inputs.
• Fig. 7.12 (Part II, §7.1, p.166). Checked against the printed page. Left pan: a bundle tagged 90 sacks, plus a 50 kg block. Right pan: a bundle tagged 60 sacks, plus a 500 kg block. Printed inputs. Nothing new is being asked; the size of the numbers is the only thing that has changed, which is exactly why it is here.
• The chapter's own instruction (Part II, §7.1, p.165). After Figs. 7.1–7.8 the reader is told to discuss the answers with classmates and give reasons. It is marked Math Talk. Treat that as content, not as a classroom aside.
• The note to the teacher (Part II, §7.1, p.166). A boxed note asks that several different strategies be encouraged and compared.

Figures to have open

• A two-pan balance whose pans can be loaded and unloaded item by item, with the beam staying level throughout a matched removal. This movement is the argument for sections 7–10; it must be built, not stated.
• A hanging mobile with a ring above, two strings below, and the ability to nest a second bar on one string. Standard schematic — redraw it; do not reproduce the book's leaf, fish, crown or sack artwork.
• A version of the hanger that visibly tips, for the counter-case in section 1.
• Sacks and labelled kilogram weights as movable objects.
• No photograph or data table from the textbook is needed.

Where this sits in the book

• NCERT Ganita Prakash, Class 7, Part II, printed Chapter 7 "Finding the Unknown", §7.1 "Find the Unknowns", bold subheading "Unknown Weights", pp.164–166 — the two opening scale pictures and Figs. 7.1–7.3 (p.164), Figs. 7.4–7.8, the Math Talk instruction, Figs. 7.9 and 7.10 with the printed hint (p.165), Fig. 7.11 with its hint, Fig. 7.12 and the note to the teacher (p.166)
• Same part, same chapter, §7.1, p.168 — the six figures the reader is asked to turn into equations
• Same part, same chapter, §7.2, p.169 — the sentence that looks back and points out that these figures were never solved by trial and error
• Forward pointer: Doing the same thing to both sides preserves equality, which turns the removal move into a statement about equations