The Land of Tens: each place is ten of the one before

What to assume they know

• What it actually takes to make one lakh — one lakh as 1 followed by five zeros, and the Indian comma placement
• Multiplication and division by 10, 100 and 1000
• Writing a number as a sum of multiples of place values, e.g. 321 as three hundreds, two tens and one unit
• Reading an expression with brackets and evaluating it

What they should be able to do

• Work out how many presses of a single fixed button are needed to reach a target, and say when the target cannot be reached at all
• Decide which numbers each of the three single-button calculators can and cannot display, and justify the answer by divisibility rather than by trying
• Write two or more different button expressions that produce the same number
• Find the fewest-press recipe for a number and write the matching expression
• Explain why the fewest-press recipe reproduces the number's digits, using the ten-of-the-one-below rule
• Predict how the press count changes when one press of a button is traded for ten presses of the next button down
• Work out the fewest presses when the available buttons do not reach the number's top place

Where it usually goes wrong

• "There is one right way to make a number." The whole of Creative Chitti exists to break this. There are many recipes — 72 of them for 321 — but finitely many; what is unique is the cheapest one.
• "The fewest presses is a rule you are told." It is a conclusion. The chapter asks the student to notice the connection between a number and its fewest-press count and does not print the answer.
• "A bigger button is always better." Only if it fits. Twelve presses of +1 beat one press of a +1000 button you cannot use.
• "Because 100 is bigger than 10, Handy Hundreds can do more." It can do strictly less: its reachable numbers are a subset of Tedious Tens'.
• "The digits and the press count are the same thing." They coincide only when the buttons reach the number's top place. With only +10,000 and +100 the count runs into the hundreds — see 65,30,000.
• "Any press count can make any number." Trading always changes the count by nine, so the reachable counts for a fixed number are spaced nine apart. That is why 999 cannot be built in exactly thirty presses.

Questions to check understanding

• "How many times must the +n button be pressed to show …?" — direct, and the chapter's own form
• Given a press count, name the number produced (the reverse direction)
• Write two different button expressions for a given number
• Write the fewest-press expression and state the press count
• Decide, with a reason, whether a given number is reachable by a given single button
• Given a restricted button set, compute the fewest presses — the case where the digit pattern fails
• Competency-style: explain why the fewest-press expression is the number's own place-value form

Examples worth working on the board

The chapter prints almost none of the answers; values marked verified are worked out here, checked against the printed page.

• Thoughtful Thousands (+1000 only), Part I, §1.2, p.5. Worked on the page: three thousand takes 3 presses. Asked and left blank: 10,000; fifty-three thousand; 90,000; one lakh; what 153 presses produces; and how many thousands make a lakh. Verified: 10, 53, 90, 100 presses; 153 presses gives 1,53,000; a hundred thousands make a lakh.
• Tedious Tens (+10 only), Part I, p.5. Asked: five hundred; 780; 1000; 3700; 10,000; one lakh; and what 435 presses gives. Verified: 50, 78, 100, 370, 1000 and 10,000 presses; 435 presses gives 4,350.
• Handy Hundreds (+100 only), Part I, pp.5–6. Asked: four hundred; 3,700; 10,000; fifty-three thousand; 90,000; 97,600; 1,00,000; what 582 presses gives; and how many hundreds make ten thousand and a lakh. Verified: 4, 37, 100, 530, 900, 976 and 1000 presses; 582 presses gives 58,200; a hundred hundreds make ten thousand and a thousand hundreds make a lakh.
• Handy Hundreds' claim (Part I, p.6). It asserts there are numbers the other two cannot show but it can, and the chapter asks the student to think it through rather than printing a verdict. Verified reasoning: every multiple of 100 is also a multiple of 10, so Tedious Tens can show everything Handy Hundreds can; the claim fails. The reverse is not symmetric — 3,700 defeats Thoughtful Thousands but not Tedious Tens.
• Creative Chitti (Part I, p.6). Buttons +1, +10, +100, +1000, +10,000, +1,00,000 and +10,00,000, drawn on the calculator in the artwork. Worked on the page: 321 as thirty-two presses of +10 and one of +1, and again as two of +100, twelve of +10 and one of +1. Verified: both give 321, at 33 presses and 15 presses; and 321 has exactly 72 recipes in all on these seven buttons, which is many but not unboundedly many.
• Two recipes for 5072 (Part I, p.6, table). Column one: fifty +100, seven +10, two +1, written as (50 × 100) + (7 × 10) + (2 × 1). Column two: three +1000, twenty +100, seventy-two +1, written as (3 × 1000) + (20 × 100) + (72 × 1). Verified: both total 5072, at 59 and 95 presses.
• Systematic Sippy's 23-press recipe for 5072 (Part I, p.7, table). Buttons +1, +10, +100, +1000, +10,000, +1,00,000. The filled table shows five +1000, zero +100, six +10, twelve +1. Verified: 5000 + 0 + 60 + 12 = 5072, and 5 + 0 + 6 + 12 = 23 presses. The chapter asks whether fewer than 23 is possible. Verified: yes — five +1000, seven +10, two +1 is 14 presses, and nothing beats it.
• Numbers set for the student, Part I, pp.6–7: 8300, 40629, 56354, 66666, 367813 — first for two different recipes each, then for the fewest presses. Verified fewest-press counts: 11, 21, 23, 30, 28.
• Chitti's puzzles (Part I, p.7). (a) Using exactly thirty presses, what are the largest and smallest 3-digit numbers reachable? (b) 997 takes 25 presses at best; can it be made in a different number of presses? Verified: 997 can be made in 34 presses as eight +100, nineteen +10, seven +1. Verified for (a): the largest is 993 and the smallest is 102. Both start from their own digits — 21 presses and 3 presses — and one trade of a bigger button for ten of the next one down adds nine, so 993 reaches 30 in one trade (eight +100, nineteen +10, three +1) and 102 reaches 30 in three (eight +10, twenty-two +1, after the +100 has already been traded away). 999 cannot be done: its own digits cost 27, and 27, 36, 45 … are the only counts it can ever have.
• The buttons that stop short (Part I, §1.6, p.20, question 7). A calculator with only +10,000 and +100. Verified: 20,800 takes 2 + 8 = 10 presses; 65,30,000 takes 653 presses, because there is no bigger button to carry into. This is the chapter's own case where the digit pattern stops working, and it belongs in section 10.

Figures to have open

• The 5072 table exactly as the chapter lays it out: button labels down the left in Indian notation (+10,00,000 down to +1), press counts in the columns. It appears twice (Part I, §1.2, pp.6 and 7) and both versions matter — the first shows two wasteful recipes, the second the 23-press one. Redraw rather than reproduce.
• Three nested rings for reachability. Standard schematic.
• A trade movement: one +1000 token becoming ten +100 tokens, with a press counter going up by nine. Standard schematic, and it is the load-bearing picture of the whole topic.
• The four calculator characters are drawn in the chapter's artwork (Part I, §1.2, pp.5–7). They are decoration for the argument, not evidence; simple original characters will do.

Where this sits in the book

• NCERT Ganita Prakash Class 7, Part I, printed Chapter 1, "Large Numbers Around Us", §1.2 "Land of Tens", pp.5–8. The section carries no numbered subheadings; the three single-button calculators, Creative Chitti and Systematic Sippy are introduced as numbered exercise items and as bold names in running text, so they can be named but not cited by number.
• The crore paragraph at the top of Part I, p.8 closes §1.2 and opens the way to §1.3; it belongs to Crores and millions: one number, two naming systems, not here.
• Part I, §1.6, p.20, question 7 (the +10,000 / +100 calculator) is the natural home of section 10 and is cited there.
• Part I, §1.2, p.7 carries a Math Talk prompt asking why the fewest-press expressions reproduce Indian place value notation. That prompt is the thesis of this topic.