PrepShorts · Teaching notes · Class 7 Mathematics · Chapter 3, Finding Common GroundPrepShorts

Chapter 3 · Finding Common Ground

Getting the HCF and the LCM out of one division ladder

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What to assume they know

What they should be able to do

  • Read a stacked-division layout and say what each row and each divisor records
  • State the condition for taking another step, and the condition for stopping
  • Carry out the layout for a pair of two- or three-digit numbers
  • Read the HCF off the finished layout, and explain why the left column gives it
  • Read the LCM off the same finished layout, and explain why the bottom row has to be included
  • Explain why dividing by a larger common factor in one step is allowed
  • Predict what happens to the two answers if the divisors are chosen in a different order, and check the prediction
  • Choose between the layout and the two separate factorisations for a given pair

Where it usually goes wrong

  • "The layout is a new rule you have to learn." It is bookkeeping for the two rules already established. Every divisor written at the left is a factor both numbers still share; the column is therefore the shared part and nothing else. Show the two separate factorisations beside the layout once, and the point is made.
  • "The divisor has to be prime." The chapter's own two characters break that and the chapter endorses them: any common factor may be taken out, and 50 and 10 are not prime. What has to be true is only that the divisor divides both numbers. Prime divisors are the safe default because you cannot overshoot with them, not because larger ones are wrong.
  • "Different choices of divisor give different answers." They give different ladders and the same two answers. This is the same stability the chapter met in Breaking a number down to primes, and why the result is unique, and section 10's interactive figure exists to make it visible.
  • "The LCM is the product of the divisors, like the HCF." Then 300 and 150 would give 150 for both. The closing row is carrying every prime that was never shared, and dropping it drops exactly those.
  • "You can stop as soon as one of the two numbers becomes prime." You stop when the pair has no common factor left. In the 300 and 150 layout the closing row is 2, 1 — a division by 3 was still available at 6, 3 even though 3 is prime.
  • "The layout is always faster." For a pair with no shared factors at all it stops immediately and tells you almost nothing beyond the HCF being 1; the LCM then still needs the product. Say when to reach for it.

Questions to check understanding

  • Find the HCF and the LCM of a given pair using the stacked layout, showing the work
  • Complete a partly filled layout and state both answers
  • Given a finished layout, say which product gives the HCF and which the LCM
  • Explain why a stated divisor may be used at a stated step
  • Compare the layout against separate factorisations for a stated pair and justify the choice
  • Word problems requiring both quantities from one pair — the closing exercise asks for HCF and LCM together, with the answers wanted as products of primes (Part II, p.63, question 3)

Examples worth working on the board

  • The unexplained layout (Part II, §3.3, p.60). Checked against — the chapter card's heading row for this block picks up only one of the ladder's four rows, so work from the printed page. It sits in the right margin beside the subheading and reads, top to bottom: divisor 2 against 84, 180; then divisor 2 against 42, 90; then divisor 3 against 21, 45; then the closing row 7, 15. No HCF and no LCM are printed for it — the reader is asked to work out how the layout was produced. The follow-up on p.61 supplies only a hint, that 84 = 2 × 2 × 3 × 7 and 180 = 2 × 2 × 3 × 15.
  • 300 and 150 (Part II, §3.3, p.61). Checked against the printed page. Divisors down the left: 2, then 5, then 5, then 3; rows 150, 75 / 30, 15 / 6, 3 and the closing row 2, 1. The book prints the HCF in factor form as 2 × 5 × 5 × 3 and the LCM as 2 × 5 × 5 × 3 × 2 × 1. Neither is evaluated on the page.
  • 630 and 770 (Part II, §3.3, p.61). Checked against the printed page. Divisors 2, then 5, then 7; rows 315, 385 / 63, 77 and the closing row 9, 11. The book prints the HCF in factor form as 2 × 5 × 7. For the LCM it prints the same layout with one extra step, a 3 taken against the row 9, 11 to give 3, 11, and the LCM in factor form as 2 × 5 × 7 × 3 × 3 × 11.
  • The two blue outlines (Part II, §3.3, p.61). Checked against the printed page, and invisible in the extracted text. Each layout is printed twice. In the HCF copies a blue curve runs down the left-hand column of divisors and stops. In the LCM copies the same curve continues round the foot of the layout to enclose the closing row as well, making an L. That single difference in the artwork is the entire lesson of sections 5 and 6 and must survive into the figure.
  • Guna's shortcut (Part II, §3.3, pp.61–62). Checked against the printed page; a cheerful illustrated boy stands beside the claim. Inputs: for 300 and 150, divide both by 50 in one step, then by 3, ending at 2, 1. The book reports the HCF as 50 × 3 and the LCM as 50 × 3 × 2 × 1, neither evaluated.
  • Anshu's version (Part II, §3.3, p.62). Checked against the printed page; a second illustrated child stands beside it. Inputs: for 630 and 770, divide both by 10 first, then by 7, ending at 9, 11. Here the book does evaluate — it reports the HCF as 70 and the LCM as 6930. Those two printed values are worth using as a check on arithmetic added here.
  • The pairs to try (Part II, §3.3, "Try This", p.62). 90 and 150; 84 and 132. Give these as inputs; nothing about them is answered in the book.

Figures to have open

  • A stacked-division layout that can be built row by row, with the divisor column and the closing row separately highlightable and separately outlineable. This is the topic's only essential figure and it does all the work.
  • The same layout able to run with non-prime divisors, so Guna's and Anshu's versions can be shown as shorter ladders reaching the same foot.
  • A side-by-side of the finished ladder and the two full prime factorisations, for section 3.
  • No numbered figure exists to cite: this chapter prints no "Fig. 3.x" label anywhere on pp.47–66 (all twenty pages checked).

Where this sits in the book

The book

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