Understand how to write prime factorization with exponents

Every whole number greater than 1 breaks into primes in exactly one way, apart from order. That is why a factor tree can start with any split and still end in the same leaves: 72 can open as 2 x 36 or as 8 x 9, and both trees finish with three 2s and two 3s.

The sheet asks for two more things. Repeated primes become powers, so 2 x 2 x 2 x 3 x 3 becomes 2³ . 3², and the factors go in order from least to greatest. The worked example, 24 = 2³ . 3, shows both rules at once.

Writing a power is 6.EE.A.1, whole-number exponents in Grade 6. The tree itself is factor work from Grade 4, put to a new use.

Start with 90, because its tree is short. Your child might split it as 9 x 10 or as 2 x 45; let them choose. When the tree stops, ask, "Is every leaf a prime?" A 9 left at the bottom means one more split.

Then comes the counting. On 108, the leaves are 2, 2, 3, 3 and 3. Ask, "How many 3s, exactly?" Kids often write 3² because they see two 3s on the last branch and forget the one higher up.

Tell your child that the dot between factors means times. A note for you: the solution page writes 90 as 2 . 5 . 3², which breaks the order rule printed at the top. In order it reads 2 . 3² . 5.

Prime factorization is a handy tool for sharing fairly. Next time you shop for party supplies, look at a pack of 24 cups: 2³ . 3 says it splits evenly among 2, 3, 4, 6, 8 or 12 kids. Ask, "Which group sizes work for 36 crackers?"

A chessboard is another good one. It has 64 squares, and 64 is 2⁶, a tree with nothing but 2s.

That is the board in The King's Chessboard by David Birch, where a wise man asks for rice doubled on every square, and the king learns too late how fast powers of 2 grow.

Name the effort: "You kept splitting until every branch ended in a prime." Next in this series, the greatest common factor sheet puts these trees to work.