Comparing two expressions by reasoning, not by evaluating

What to assume they know

• Writing a situation as an expression before computing it — an expression names a number; that number is its value
• Ordering whole numbers, and the signs <, >, =
• Addition and subtraction of three-digit numbers, at least well enough to check
• Class 6 experience of "one more" and "one less" on a number line

What they should be able to do

• Place <, > or = between two expressions and explain that the sign is a statement about their values
• Compare two sums by comparing their corresponding parts, without evaluating either side
• Compare two differences the same way, correctly reversing the effect of a change that lands on the amount being subtracted
• Retell a numeric comparison as a story about two people, and read the answer off the story
• Read a labelled bar picture of a comparison and say which feature of the picture carries the conclusion
• Judge when reasoning is genuinely quicker and when it is honest to just compute
• Arrange several expressions in order of value, choosing the cheapest route to each comparison

Where it usually goes wrong

• "To compare two expressions you must evaluate both." The chapter's whole point in Examples 2 and 3 is that a situation can settle the comparison first. Evaluating is a fallback, not the method.
• "The side with the bigger first number is bigger." 1023 + 125 starts ahead and finishes behind. Corrected by the bar picture on p.25.
• "Adding 1 somewhere always adds 1 to the value." True when the 1 lands on something being added. In 113 – 25 the extra 1 lands on the amount removed and pulls the value down, which is exactly why the two changes in Example 3 cancel.
• "Equal must mean somebody worked both sides out." 113 – 25 = 112 – 24 is settled by matching the changes; neither total is needed.
• "The longer or more complicated-looking side is the larger." 13 – 2 and 4 × 3 are the counter-case printed on p.25.
• "Reasoning always beats calculating." Overclaiming here is the trap. In the ordering task on p.25, three of the five expressions have nothing to line up against, and computing them is the sensible move. The chapter asks whether the comparison can be done without complicated calculation; it does not promise it.

Questions to check understanding

• Put <, > or = between two given expressions and justify the choice without evaluating (Part I, §2.1, p.26 — the instruction to explain the thinking is part of the question)
• Fill a blank so that two expressions are equal (Part I, §2.1, p.25)
• Arrange a set of expressions in increasing order of value (Part I, §2.1, p.25)
• Given a pair that differs in two places, say which change dominates
• The same competence returns much later in the chapter as a longer comparison set, once brackets and the distributive property are available (Part I, p.43)

Examples worth working on the board

Where a comparison is printed without its values, keep it that way — supplying the totals defeats the exercise.

• The two printed comparisons (Part I, §2.1, pp.24–25): 10 + 2 set against 7 + 1, and 13 – 2 set against 4 × 3. The chapter justifies the first one by naming both values; the second it simply asserts. The second is the more useful of the two, because the sides do not resemble each other.
• Example 2 — Raja and Joy adding (p.25). Compare 1023 + 125 with 1022 + 128. The story: Raja held 1023 marbles and received 125 more; Joy held 1022 and received 128. Inputs for the reasoning, taken off the printed bar picture: 1023 is 1022 and one more; 128 is 125 and three more. The chapter states the conclusion as a < between the two expressions, and says in words that Joy ends two ahead.
• Example 3 — Raja and Joy subtracting (p.25). Compare 113 – 25 with 112 – 24. The story: Raja held 113 and lost 25; Joy held 112 and lost 24. Inputs: Raja starts one ahead, and also loses one more. The chapter's conclusion is =, and this is the case worth dwelling on, because two changes that both look like "one more" work against each other.
• The five-pair drill (p.26). The chapter asks for a sign in each box and, crucially, for the thinking behind it. The pairs, with the shift in each part (do not pre-compute the totals when explaining it):
  • 245 + 289 against 246 + 285 — first part up 1, second part down 4
  • 273 – 145 against 272 – 144 — starting amount down 1, removed amount down 1
  • 364 + 587 against 363 + 589 — first part down 1, second part up 2
  • 124 + 245 against 129 + 245 — first part up 5, second part unchanged
  • 213 – 77 against 214 – 76 — starting amount up 1, removed amount down 1
• Fill-in pairs (p.25): 13 + 4 = ___ + 6; 22 + ___ = 6 × 5; 8 × ___ = 64 ÷ 2; 34 – ___ = 25. Each is an equality between two expressions rather than between an expression and a numeral.
• Ordering task (p.25): put 67 – 19, 67 – 20, 35 + 25, 5 × 11 and 120 ÷ 3 into increasing order. The first two differ only in the removed amount, so that pair is settled by reasoning; the rest need real values. Use this to make section 10's point honestly.

Figures to have open

• The Example 2 bar picture (Part I, p.25). Two labelled rows. Raja's row: a long bar marked 1022 with a single unit chip beside it, and a shorter bar marked 125. Joy's row: a bar marked 1022, a bar marked 125, and three separate unit chips. Redraw it — do not reproduce the printed art — but keep the decomposition, because that decomposition is the argument.
• The Example 3 bar picture (Part I, p.25). Two rows again. Raja: a bar marked 112 with a unit chip beside it, and a labelled removal bracket spanning 24 of the bar plus that chip. Joy: a bar marked 112 with a removal bracket spanning 24. The visual point is that Raja's bracket reaches one unit further than Joy's, exactly matching the extra unit he started with. Both pictures were checked against p.25.
• A shift-tracker for section 9 — an added device: two stacked expressions with a signed arrow over each part that changed.

Where this sits in the book

• NCERT Ganita Prakash, Class 7, Part I, printed Chapter 2 "Arithmetic Expressions", §2.1 "Simple Expressions", the unnumbered subheading "Comparing Expressions" — pp.24–26. The subheading starts on p.24; Examples 2 and 3 and both bar pictures are on p.25; the five-pair drill is at the top of p.26.
• Chapter SUMMARY, Part I, p.44 — the second bullet is about comparing by reasoning rather than by evaluating.
• Forward pointer: Part I, printed Chapter 2, p.43, where the comparison exercises return in a form that needs brackets and terms.