Hop Back on the Number Line: What Is Left?

Three number lines stretch across this 3rd grade page, one per card, each starting at 0 and marked every unit with the fives labelled. A teal dot sits on each line at the dividend — 15 on a line to 20, 22 on a line to 25, 10 on a line to 15 — and the division is printed beneath: 15 ÷ 4, 22 ÷ 3, 10 ÷ 3, each followed by a box and an R box. No hops are drawn. The child starts at the dot and draws hops back of the divisor's size — fours from 15 — until a whole hop would go past 0, then writes how many hops they drew and the number they landed on.

The number line makes the remainder a place rather than a count: hopping back three fours from 15 lands on 3, and 3 is where you stop because a fourth hop would need four more than you have. That is a different picture from rings on a pile, and for some children a clearer one, because the leftover is visible as the gap between the landing point and 0. Drawing the hops themselves, rather than reading printed ones, is what makes the page a division page — the child chooses the hop size from the divisor and keeps it constant.

One sheet, three lines, six boxes; the lines are long enough for hops of 2 to 5 without crowding. This is the measurement face of the division-with-remainders family, the dividends kept between 10 and 30 so the hops stay countable. Use it beside the ring-the-groups page to show the two pictures of one division, and before the practice-rows page.