Which Way, and How Far Operations with Integers

Rakesh thinks of two numbers whose sum is 25 and whose difference is 11, and finding them by guessing and checking builds exactly the intuition integer arithmetic needs: adjusting one guess up and the other down by the same amount keeps the sum fixed while changing the difference. The same puzzle style extends naturally into negative territory -- a sum of 0 and a difference of -10 has a perfectly good answer (-5 and 5), just as valid as any pair using only positive numbers.

A coin on a number line, struck rightward by some amount then struck again (rightward or leftward) by another amount, always ends at a position found by simply adding the two movements together, P = a + b, treating a rightward strike as positive and a leftward strike as negative. This single formula quietly covers all four possible sign combinations at once (positive-then-positive, positive-then-negative, negative-then-positive, negative-then-negative), since the direction is already built into whether each movement counts as positive or negative.

The zero-pair token model from Class 6 (green=positive, red=negative, a matching pair cancels to nothing) extends smoothly to trickier subtractions: (+7) - (+18) needs 11 more greens removed than are actually present, solved by first adding 11 zero-pairs (changing nothing about the pile's true value), then removing all 18 greens, leaving 11 reds, confirming (+7)-(+18) = -11.

Using bags of tokens, 4x2 means placing 4 bags of 2 positive tokens each, giving 8 positives; 4x(-2) means placing 4 bags of 2 negative tokens each, giving 8 negatives. But (-4)x2 means something genuinely different: REMOVING 4 bags of 2 positive tokens from an empty starting pile, which needs zero-pairs added first before any removal is possible, and ends up leaving 8 negatives behind. Finally, (-4)x(-2) means removing 4 bags of 2 negative tokens, again needing zero-pairs first, and ends up leaving 8 positives. Laying out a full multiplication table for a fixed multiplicand across positive and negative multipliers makes the pattern undeniable: same signs multiply to a positive result, different signs multiply to a negative result, every single time.

An exam awards +5 for each correct answer and -2 for each wrong one; with 30 correct and 20 wrong answers out of 50, the total is 30x5 + 20x(-2) = 150-40 = 110 marks, integer multiplication doing real work. A mining elevator descending at a steady rate for a fixed time, worked two different ways (finding the total distance first, or finding the combined speed-and-direction first), always lands on the same final position, confirming multiplication of signed quantities is consistent regardless of which of two equally valid methods is used. Division of integers is simply multiplication asked in reverse -- 36 divided by (-18) asks what integer times -18 gives 36, and following the same same-sign/different-sign pattern already established for multiplication answers it directly: -2.

Multiplying several integers together in a different grouping or a different order, like 5x(-3)x4 rearranged as (-3)x4x5 or 4x5x(-3), always lands on the exact same final answer, -60, confirming that the familiar reordering and regrouping freedoms from whole-number arithmetic carry over completely intact into integers. The same holds for distributing a multiplication across a sum or difference: multiplying a token rectangle's whole length by its height gives the same total as splitting the rectangle into two pieces, multiplying each piece separately, and adding the two partial results together.