Factorise and regroup instead of multiplying head-on

What to assume they know

• The Land of Tens: each place is ten of the one before — that each place is worth ten of the one below, so multiplying by ten shifts every digit one place
• Multiplying and dividing whole numbers by 10, 100 and 1000
• Dividing a 3- or 4-digit number by 2, 4 or 8 mentally
• Reading a product written with brackets, and knowing that the order of the factors may be changed

What they should be able to do

• Rewrite 5 as ten halved, 25 as a hundred quartered and 125 as a thousand divided by eight, and use each to compute a product in one line
• Explain, in terms of what multiplication and division mean, why halving and then multiplying by ten is the same as multiplying by five
• Regroup a product of three or four factors so that the easy pairs come together
• Decide, given a factor, whether the shortcut applies at all
• State which whole numbers have a shortcut of this kind and why
• Recognise when the required division does not come out whole, and say what that costs
• Work backwards from a stated product to a pair of factors that would produce it

Where it usually goes wrong

• "This is a special rule for 5." It is a rule about powers of ten. Once a student sees 25 and 125 done the same way, the pattern is the point.
• "Dividing by 2 makes the answer smaller, so the trick must lose something." Halving and then multiplying by ten is multiplying by five. The two moves are one move written in two steps, not an approximation.
• "You can do this with any number." 3, 7, 11, 13 have no such rewriting. Testing 3 in front of the class is worth a section.
• "Multiplying by 10 adds a zero." It shifts every digit one place; the zero is what fills the vacated units place. The distinction matters the moment decimals appear in a later class.
• "You must multiply the factors in the order written." The whole of 125 × 40 × 8 × 25 depends on refusing to. Pairing 125 with 8 and 40 with 25 turns four multiplications into one.
• "The shortcut fails if the division is not exact." It does not fail; it becomes ordinary work. 823 × 25 is still 82,300 ÷ 4.

Questions to check understanding

• "Find a quick way to calculate …" on a product with a factor of 5, 25, 50 or 125 — the chapter's own form
• Compute a product mentally and write the regrouping that made it easy
• Fill a blank pair of factors for a given product
• Explain why halving and multiplying by ten is the same as multiplying by five
• Decide whether a given factor admits this shortcut, with a reason
• Regroup a product of four factors to make two round pairs
• Competency-style: given two students' methods for the same product, say which is quicker and why

Examples worth working on the board

Values marked verified are worked out here against the printed page; the chapter prints the two openers worked and leaves the rest blank.

• Roxie's line (Part I, §1.5, p.13, worked on the page). 116 × 5 is written as 116 × (10 ÷ 2), the 116 and the 2 cancelled with 58 written above, giving 58 × 10 = 580. The cancellation is set as a struck-through fraction, so the visual move is divide first, then shift.
• Estu's line (Part I, §1.5, p.14, worked on the page). 824 × 25 is written as 824 × (100 ÷ 4), the 824 and the 4 cancelled with 206 written above, giving 20,600.
• The Math Talk prompt (Part I, §1.5, p.14): explain, using what multiplication and division mean, why multiplying by five is the same as halving and multiplying by ten. Verified reasoning: ten halved is five, so the two instructions name the same factor; nothing about 116 was used.
• Quick products set for the student (Part I, §1.5, p.14). 2 × 1768 × 50; 72 × 125, with the chapter's own hint that 125 is a thousand divided by eight; and 125 × 40 × 8 × 25. Verified: 1,76,800 (pair the 2 with the 50 to make 100); 9,000 (72 ÷ 8 = 9, then × 1000); 10,00,000 (pair 125 with 8 to make 1000, and 40 with 25 to make another 1000).
• Five more (Part I, §1.5, p.14). 25 × 12; 25 × 240; 250 × 120; 2500 × 12; and a blank pair whose product is 120000000. Verified: 300; 6,000; 30,000; 30,000. The blank has many answers — 1200 × 1,00,000 and 12 × 1,00,00,000 both work — and the chapter leaves it open on purpose.
• The shortcut table to build in section 6. Verified: 5 = 10 ÷ 2; 4 = 100 ÷ 25 but more usefully 25 = 100 ÷ 4; 8 = 1000 ÷ 125 and 125 = 1000 ÷ 8; 50 = 100 ÷ 2; 250 = 1000 ÷ 4; 2500 = 10,000 ÷ 4. Each row is a power of ten over a small whole number.
• The numbers with no shortcut. Verified: 3 divides no power of ten, because 10, 100, 1000 and every later one leave remainder 1 when divided by 3; the same holds for 7 and 11. The whole numbers that do divide a power of ten are exactly those made only of twos and fives — 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125 and so on.
• When it is not tidy. Verified, an added example: 823 × 25 becomes 82,300 ÷ 4 = 20,575. The shortcut still works; it is just no longer a mental calculation, because 823 does not divide by 4. What makes the chapter's cases pleasant is that 116, 824 and 72 divide exactly by 2, 4 and 8.

Figures to have open

• The two worked lines exactly as the chapter sets them, with the cancellation strokes (Part I, §1.5, pp.13–14). The struck-through fraction is what carries the method; redraw it rather than reproducing the printed art.
• A place-value strip showing a shift by one, two and three places. Standard schematic; this is the picture behind "multiplying by ten is free".
• The family tree of powers of ten and their divisors. Standard schematic; the chapter prints no such figure and it is the topic's generalisation.
• No photograph or data figure 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.13–14, under the unnumbered subheading "A Multiplication Shortcut". §1.5 opens on p.13 with two lines of framing and then goes straight into this subheading; neither the subheading nor the Math Talk prompt on p.14 carries a number.
• Part I, SUMMARY, p.22 is where the chapter names the move — factorising and regrouping to make a multiplication simpler — so the two key terms factorise and regroup are cited there rather than to §1.5.
• "How Long is the Product?" (Part I, §1.5, pp.14–15) and "Fascinating Facts about Large Numbers" (Part I, §1.5, pp.16–18) are the other two unnumbered subheadings of §1.5 and belong to Predicting how many digits a product will have, before multiplying and Fascinating Facts about Large Numbers.