Problem
The relationship allows either side, so the number is a factor of 48 or a multiple of 48.
Factor and multiple are the two sides of the same divisibility coin, so checking both sides covers every possibility.
4.OA.B.4Identify SubproblemsA number in a factor-multiple relationship with 48 is either a factor of 48 or a multiple of 48.
The relationship names one number the factor and the other the multiple, and the mystery number is allowed to sit on either side.
Saying one number is a factor of another means the second is made of a whole number of copies of the first, so the pair always has a smaller side and a larger side.
The two cases can overlap, so 48 shows up on both lists and still counts as only one number.
Counting how many numbers work means matching each distinct number to one entry in the count, so a repeat adds nothing new.
All factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Keeping only the two-digit ones gives 12, 16, 24, and 48.
Pairing factors (each small factor with its partner) makes sure no factor is missed.
4.OA.B.4Make A Systematic ListThe multiples of 48 are 48, 96, 144, ... Only 48 and 96 are two-digit numbers.
Counting by 48 quickly leaves the two-digit range, so the list of two-digit multiples is short.
4.OA.B.4Make A Systematic ListMerging {12, 16, 24, 48} with {48, 96} and counting 48 only once leaves 5 numbers.
A two-digit value should be counted only once even if it satisfies the relationship in both ways.
4.OA.B.4Make A Systematic ListFactor or multiple? List both, then count each number just once.