Reasoning · Grade 2-2 Conditions and Numbers
Problem
Walking the digits sorts them and also catches the ones fitting neither.
Comparing neighboring digits with ＞ or ＜ is the same digit-comparison skill used to order two-digit numbers.
1.NBT.B.3Look For A PatternWalking the digits, 819 is neither, 279 is ascending and 7321 is descending.
Walking left to right and comparing each digit to the next tells you instantly whether it climbs, falls, or does neither.
1.NBT.B.3Make A Systematic ListFalling from the top digits gives 9876.
A bigger number wants big digits in the high places, and 9 8 7 6 already falls — so it is automatically descending.
2.NBT.A.4Guess And CheckNeeding three smaller digits beneath makes the smallest 3210.
You need four different digits getting smaller, so the lowest possible top digit is 3, sitting over 2, 1, 0.
2.NBT.A.4Guess And CheckChoosing which card to drop gives 3568, 1568, 1368, 1358, 1356 — five.
Sorting four chosen digits upward fixes the number, so counting the numbers is just counting which single card to drop.
1.NBT.B.3Make A Systematic ListOnce four cards are chosen, sorting them upward fixes the number completely, so counting ascending numbers is just counting which card to leave out.
Four different digits can be put in increasing order in exactly one way, so each set of four cards names one ascending number and no more.
Each ascending number comes from exactly one set of four cards, so numbers and card-sets match up one for one.
Just walk along the digits — if they always climb it's ascending, always fall it's descending, and to count ascending numbers you only pick which digits to use, because they sort themselves!