Reasoning · Grade 5-1 Primes and Divisors
Problem
In the examples the circle count follows the prime structure.
The examples do the explaining: once you see that 'right' always means one more 2 and 'up' always means one more 3, the picture stops being decoration and becomes a filing cabinet for divisors.
4.OA.B.4Solve An Easier Related Problem36 is two 2s and two 3s.
Repeated division by the smallest prime is the standard factor-tree routine, and writing the repeats as exponents is just shorthand for 'how many times this prime appears'.
6.EE.A.1Look For A PatternSo the diagram is 3 by 3.
Each row is the row below it multiplied by 3, so after filling the bottom row the rest is doubling-and-tripling arithmetic a fourth grader can do in their head.
4.NBT.B.5Draw A DiagramThe divisors number 3 × 3 = 9.
The '+1' is the easy thing to forget: using a prime zero times is a legitimate choice, so an exponent of 2 offers three options, not two. The grid makes that visible — the bottom row starts at 1, not at 2.
6.EE.A.1Make A Systematic ListThere are exactly nine divisors because each prime's exponent can be chosen independently, zero included.
A number breaks into primes in only one way, so a divisor is fully named by how many copies of each prime it takes.
How many twos a divisor takes puts no limit on how many threes it takes, so the two counts of choices multiply.
150 is one 2, one 3 and two 5s.
The exponents literally draw the box for you: an exponent of 1 gives a side 2 circles long, an exponent of 2 gives a side 3 circles long.
6.EE.A.1Look For A PatternThat makes a box of 12 circles.
Each level is the level below multiplied by 5, so once the bottom four are placed the other eight are just two rounds of multiplying by 5.
4.NBT.B.5Draw A DiagramThe divisors number 2 × 2 × 3 = 12.
Multiplying the choices is the same 'one from each column' counting used for outfit combinations; here the columns are the primes and the items are how many copies of that prime to use.
6.EE.A.1Make A Systematic ListListing them out also gives 9 and 12.
Listing divisors by testing each number is something a fourth grader can already do; matching that list against the picture proves the shortcut is trustworthy.
4.OA.B.4Make A Systematic ListEach prime is one direction to walk in, and an exponent of e lets you take 0, 1, ... , e steps — so multiply the (e+1)s and you have counted every divisor without listing one.