Predicting how many digits a product will have, before multiplying

What to assume they know

• Factorise and regroup instead of multiplying head-on — regrouping a product, and multiplying by powers of ten
• The Land of Tens: each place is ten of the one before — the smallest and largest number with a given count of digits
• Multiplying two 3- or 4-digit numbers by hand
• Comparing large numbers, and counting the digits in a written number

What they should be able to do

• Evaluate a short chain of patterned products and describe the pattern in the answers
• State what a pattern in four examples does and does not establish
• Give the smallest and largest number having a stated count of digits
• Bound the product of two numbers by multiplying the extreme cases, and read the digit count off the bounds
• State how many digits the product of an m-digit and an n-digit number can have, and show that no other count is possible
• Decide whether a stated combination of digit counts can occur, giving a reason rather than an example search
• Say what the digit-count rule cannot tell you

Where it usually goes wrong

• "Four examples that fit means the pattern is true." Box four's neat reading breaks at 105 × 105. The chapter puts patterns immediately before an argument for exactly this reason.
• "You would have to try all the pairs." You would have to try 8,100. Roxie tries two. Naming the number 8,100 is what makes her move feel like a gain.
• "The largest product of two 2-digit numbers is 100 × 100." It is 99 × 99 = 9,801. Roxie multiplies 100 × 100 deliberately because it is easy and safely above every real case — that is the whole idea of a ceiling, and students read it as an error unless it is spelled out.
• "The product has as many digits as the two factors together." It has that many or one fewer. Both really occur.
• "The rule tells you the product." It tells you its length. 12-digit × 13-digit gives 24 or 25 digits and nothing more.
• "Multiplying always adds digits." 1-digit × 1-digit can stay at one digit: 2 × 3 = 6.
• "3-digit × 3-digit could be 4 digits if the numbers are small enough." The smallest they can be is 100 each, and that already gives five digits.

Questions to check understanding

• Evaluate a patterned chain and extend it by one line
• State the smallest and largest number with a given digit count
• Predict the digit count of a product without multiplying, and justify it
• Decide whether a stated digit-count combination is possible, with a reason
• Fill blank rows of a digit-count table
• Given a claim about all numbers of some kind, decide whether checking examples could establish it
• Competency-style: explain why testing two products settles a claim about thousands of them

Examples worth working on the board

The four boxes are printed with the right-hand sides blank; values marked verified are worked out here.

• Box one (Part I, §1.5, p.14): 11 × 11, 111 × 111, 1111 × 1111. Verified: 121; 12,321; 12,34,321. Extending, 11111 × 11111 = 12,34,54,321.
• Box two (Part I, p.14): 3 × 5, 33 × 35, 333 × 335. Verified: 15; 1,155; 1,11,555. Extending, 3333 × 3335 = 1,11,15,555.
• Box three (Part I, p.14): 66 × 61, 666 × 661, 6666 × 6661. Verified: 4,026; 4,40,226; 4,44,02,226 — fours, then a zero, then twos, then a six, one more of each block at every step.
• Box four (Part I, p.14): 101 × 101, 102 × 102, 103 × 103. Verified: 10,201; 10,404; 10,609. Note for section 3: the tempting reading of these — 1, then 0, then twice the small number, then its square — survives 104 × 104 = 10,816 but visibly breaks at 105 × 105 = 11,025, because the square runs to three digits and carries. This is the chapter's own material turned into the warning the section needs.
• Roxie's claim and her reasoning (Part I, §1.5, p.15). Multiply any two 2-digit numbers, she claims, and the answer must run to either three digits or four — never fewer, never more. Her argument, printed on the page: the smallest case is 10 × 10, so every answer clears 100; and 100 × 100, the two smallest 3-digit numbers multiplied, gives 10,000, so every answer stays under 10,000.
• How many cases she avoided. Verified: there are 90 two-digit numbers, so 8,100 ordered pairs, or 4,095 pairs once you stop counting a × b and b × a separately. Two multiplications replace all of them.
• The two follow-up questions (Part I, §1.5, p.15). Can a 3-digit number times a 3-digit number give a 4-digit answer? Can a 4-digit times a 2-digit give a 5-digit answer? Verified: no to the first — the smallest such product is 100 × 100 = 10,000, already five digits. Yes to the second — 1000 × 10 = 10,000 is five digits, and the largest, 9999 × 99 = 9,89,901, is six.
• The multiplication statements (Part I, §1.5, p.15, table). Printed filled: 1-digit × 1-digit gives 1 or 2; 2 × 1 gives 2 or 3; 2 × 2 gives 3 or 4; 3 × 3 gives 5 or 6. Printed blank: 5-digit × 5-digit, 8-digit × 3-digit, 12-digit × 13-digit. Verified: 9 or 10; 10 or 11; 24 or 25.
• The general rule. Verified: an m-digit number is at least 1 followed by m − 1 zeros and less than 1 followed by m zeros. Multiplying the two floors gives 1 followed by m + n − 2 zeros, so the product has at least m + n − 1 digits; multiplying the two ceilings gives less than 1 followed by m + n zeros, so it has at most m + n. Both happen: 10 × 10 = 100 is the short case and 99 × 99 = 9,801 the long one.

Figures to have open

• The four pattern boxes as the chapter lays them out, two above and two below (Part I, §1.5, p.14). Redraw; the layout matters because the explanation fills them in.
• The multiplication-statement table (Part I, §1.5, p.15), with its four filled rows and three blank ones. Redraw; this is the topic's spine.
• A number line, or a shaded band, with 100 at one end and 10,000 at the other and every two-digit product shown falling inside it. This is the picture the chapter does not print and the argument most needs. Standard schematic.
• No photograph is required.

Where this sits in the book

• NCERT Ganita Prakash Class 7, Part I, printed Chapter 1, "Large Numbers Around Us", §1.5 "Patterns in Products", pp.14–15, under the unnumbered subheading "How Long is the Product?". The subheading carries no number.
• Roxie's bounding argument is printed as running text beside her portrait on Part I, p.15.
• The three unanswered rows of the multiplication-statement table are on Part I, p.15.
• The neighbouring subheadings of §1.5 are "A Multiplication Shortcut" (Part I, pp.13–14), which is Factorise and regroup instead of multiplying head-on, and "Fascinating Facts about Large Numbers" (Part I, pp.16–18), which the spine marks as an activity set rather than an explanation.