Reasoning · Grade 4-1 Arithmetic Puzzles
Problem
The second row is the product with the tens digit 3.
Naming what each row is before touching any box is the same habit as labelling the parts of a word problem — it turns a wall of squares into two ordinary multiplications.
4.NBT.B.5Make A Systematic ListThat row puts the multiplicand between 602 and 632.
Turning a range of products back into a range of factors is just dividing the two ends, and counting by tens from 602 to 632 gives a list a fourth grader can hold in their head.
4.NBT.B.6Make A Systematic ListA four-digit first row forces the ones digit 6.
Only the ones digits matter for the last digit of a product, so a single row of the two times table tests all ten possibilities at once.
4.NBT.B.5Eliminate PossibilitiesTesting candidates, only 632 fits the digits.
With only four candidates left, testing them all is quick, and column addition lets one single digit of the answer decide between them.
4.NBT.B.5Guess And CheckCompleted, it reads 632 × 36 = 22752.
The shifted row is worth ten times what it looks like, because every digit has slid one place to the left — that is exactly why the two rows can be added straight down the columns.
4.NBT.A.1Guess And CheckThe shifted row is worth ten times what it looks like, which is exactly why the two rows can be added straight down the columns.
Every digit in that row has slid one place to the left, and each place is worth ten of the place to its right.
The whole product is the ones part and the tens part multiplied separately and then added, so the two rows really are the two pieces of one calculation.
Start with the row that shows the most digits — one full row can trap the hidden number between two bounds, and then only a few tries are left!