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Chapter 7 · Finding the Unknown

An equation is a claim of equality, not an instruction to compute

Teaching notesNCERT10 min

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10 min.

What to assume they know

  • Writing a situation as an expression before computing it — writing a situation as an arithmetic expression before computing it
  • Turning a pattern into a formula that predicts — turning a pattern into a formula that predicts, and reading an expression like 2n + 1; Part I, §4.5 is where the triangle-strip matchstick sequence itself was met
  • Adding and subtracting negative numbers, and the meaning of −(−5)
  • Reading a term-by-term count off a picture (the matchstick sequence is revisited from an earlier chapter of Part I)
  • Multiplication tables to the point where 2 × 49 is immediate

What they should be able to do

  • Read a level hanging bar as an assertion that two collections agree, not as a sum to be carried out
  • State what makes a written line an equation rather than an expression
  • Identify which side of a given equation is its LHS and which its RHS
  • Turn a described unknown quantity into a named letter-number and write the equality it satisfies
  • Derive the expression 2n + 1 for the nth matchstick arrangement from the first few counts
  • Explain what "solving" an equation asks for, and why it is a different job from evaluating an expression
  • Decide whether a stated stick count can be reached at all, and say what makes it impossible when it is
  • Justify why an equation is worth writing down before any method for solving it is available

Where it usually goes wrong

  • **"The equals sign means now write the answer."** The whole opening is built against this. A level bar issues no instruction; it reports a state of affairs. Section 2 should stop and name the shift explicitly, because every later step depends on it.
  • "An expression and an equation are the same kind of thing." 2n + 1 names a number once you fix n. 2n + 1 = 99 asserts something that is true for exactly one n and false for every other. The chapter puts the definition immediately after the sequence work so the contrast is fresh.
  • "The answer always belongs on the right." 20 = y − 3 is printed on p.167 for precisely this reason, and Example 16 later ends on 100 = x. Read equations in both directions from the start.
  • "A letter is a mystery box you are supposed to guess." The letter is a name, chosen by the person writing the problem — e for the egg, y for the ring, n for the position. Naming is a decision, not a puzzle.
  • "If I cannot solve it, there is no point writing it down." The exact reverse of this chapter's move. Jasmine's equation is written on p.167 and no method for it exists until p.168. Writing it is what makes a method worth having.
  • "2n + 1 = 200 just has no answer." Too flat. Half of 199 is a perfectly good number; what fails is the demand that a position in this sequence be a whole number. Keep the reason attached to the claim.

Questions to check understanding

  • Given a described situation, name the unknown with a letter and write the equation it satisfies
  • Name the two sides of a given equation using the book's own labels
  • Say whether a given line is an expression or an equation, and why
  • Given the first few terms of a stick or tile sequence, write the expression for position n
  • Given a target count, decide whether any position in a stated sequence reaches it, and justify the answer
  • Check a proposed value by substituting it and comparing the two sides
  • The chapter's own instruction after the opening figures is to argue for your answer to a classmate rather than to check it against a key (Part II, §7.1, p.165) — reasoning-out-loud items of that shape are what this topic seeds

Examples worth working on the board

  • The two opening pictures (Part II, §7.1, p.164). Checked against the printed page. Left picture: a straight horizontal bar hanging from a ring marked 4, with a character marked 2 gripping each end — the bar is dead level. Right picture: a bar hanging from a ring marked 7, with a character marked 4 on the left and one marked 3 on the right — and this bar visibly tilts, low on the 4 side. Two facts are being taught at once by picture alone: the ringed number is the whole weight hanging below, and the bar is level exactly when the two sides agree. Neither fact is written out in words on that page.
  • Fig. 7.1 (Part II, §7.1, p.164). Checked against the printed page. Ring marked 16. One string carries a leaf, a small dark bud and a second leaf; the other string carries a flower. The legend below gives leaf = 3 and leaves two blanks, bud and flower.
  • Fig. 7.6 (Part II, §7.1, p.165). Checked against the printed page. A bar held level in someone's hand, no ring, no total. Three slices of bread hang on the left, two fried eggs on the right, and the legend gives bread = 2. This is the figure the chapter later converts into symbols, and its point is that the level bar alone is enough — no total needed.
  • Framing Fig. 7.6 (Part II, §7.1, p.168). The chapter names the weight of one egg e, sets three twos against e + e, and writes the equality both ways round, ending at 2e = 6. Use both orderings; the chapter deliberately writes the sum on the right first.
  • Framing Fig. 7.7 (Part II, §7.1, p.168). Inputs: the star weighs 4; the ring-shaped item is called y; one side totals 16 and the other is 4 + 2y. The printed equation is 4 + 2y = 16.
  • The matchstick sequence (Part II, §7.1, p.166). Checked against the printed page. Four arrangements of matchsticks drawn in a row and labelled 1, 2, 3, 4. Each is a strip of triangles sharing sides: one triangle, then two, then three, then four. Position 1 uses 3 sticks, position 2 uses 5, position 3 uses 7.
  • The counts and the expression (Part II, §7.1, p.167). The chapter writes each count in the same shape — 2 × (1) + 1, 2 × (2) + 1, 2 × (3) + 1 — before generalising to 2n + 1. Keep that layout; the pattern is visible only because the arithmetic is left undone.
  • Jasmine's 99 sticks (Part II, §7.1, p.167). Input: 2n + 1 = 99. The chapter does not solve it on this page; it asks the reader for ways to get n. Leave it open here.
  • The 200-stick question (Part II, §7.1, p.167). Input: the same sequence, and a demand for exactly 200 sticks. State the condition precisely — every arrangement in this sequence uses an odd number of sticks, so no position in it uses 200. Do not say "2n + 1 = 200 has no solution" without that qualification; the arithmetic sentence has a perfectly good half-integer answer, and it is the sequence, not the algebra, that rules it out.
  • The LHS/RHS callout (Part II, §7.1, p.167). Checked against the printed page. Two labelled boxes with arrows pointing down at the two halves of 2n + 1 = 99.
  • Further printed equations (Part II, §7.1, p.167). The chapter offers four more as examples of the form: 3x + 4 = 7; 20 = y − 3; a ÷ 3 = 50 (set as a fraction); 2z + 4 = 5z − 14. Use at least the second and the fourth — one has the lone number on the left, the other has letters on both sides, and both break the "answer goes on the right" habit.

Figures to have open

  • A hanging bar with a ring above it, redrawable level and tilted, with numbers settable on the ring and on each side. Standard schematic — redraw it; do not lift the book's cartoon characters.
  • A mobile with one branch carrying a chain of three objects and the other carrying a single object, matching the layout of Fig. 7.1. Standard schematic.
  • The strip-of-triangles matchstick sequence at positions 1 to 4, with individual sticks countable. Standard schematic, but the shared-side structure is load-bearing: the arrangements must share edges, or the count stops being 2n + 1.
  • An equation with its two sides highlighted separately, for the LHS/RHS naming.
  • 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", pp.164–168 — the opening scale pictures and Figs. 7.1–7.3 (p.164), Figs. 7.6–7.8 (p.165), the bold subheading "Matchstick Pattern" and the sequence (p.166), the derivation of 2n + 1, Jasmine's question, the definition of equation, the further examples and the LHS/RHS callout (p.167), and the framing of Figs. 7.6 and 7.7 as equations (p.168)
  • Same part, same chapter, SUMMARY, p.190, bullets 1 and 3
  • Backward pointer: the matchstick sequence is stated on p.166 to have been met in an earlier chapter; that is Part I, printed Chapter 4, §4.5 "Pick Patterns and Reveal Relationships", where the same strip of triangles is counted and the count is written as 3 + 2 × (y − 1) — in this spine, Turning a pattern into a formula that predicts
  • Forward pointer: Doing the same thing to both sides preserves equality, where 2n + 1 = 99 is finally solved

The book

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