Mission 2 · Unit 1

Prime Factorization

6.NS.B.4 · Unit 1
Today's objective: Break numbers into prime factors and use them to find GCF and LCM.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The school band and the school choir are performing together at the spring concert. The band plays a repeating pattern every 18 beats and the choir claps every 24 beats. The music director needs to know: When will the band pattern and the choir clap land on the exact same beat for the first time? Your team will use prime factorization and factor trees to find the LCM and answer this question.

Spring Concert Beat Sync When do the band and choir land on the same beat? Factor Tree: 18 18 2 9 3 3 18 = 2 x 3 x 3 18 = 2 x 3² Factor Tree: 24 24 4 6 2 2 2 3 24 = 2 x 2 x 2 x 3 24 = 2³ x 3 LCM = ? beats

Team Roles

Facilitator Reads the concert beat problem aloud, makes sure every person builds at least one factor tree, and tracks the phase timer.
Model Builder Draws factor trees for 18 and 24, writes the prime factorization using exponents, and builds the Venn diagram of shared primes.
Precision Checker Multiplies the prime factors back to check each tree, verifies the LCM by confirming both 18 and 24 divide evenly into it.
Reporter Prepares the defense: explains the factor trees, shows how LCM was found, and describes one revision the team made.

Investigation

The Problem

The school band repeats its pattern every 18 beats. The choir claps every 24 beats. Both start at beat 0.

Your tasks:

  1. Build a factor tree for 18 and write its prime factorization.
  2. Build a factor tree for 24 and write its prime factorization.
  3. Use the prime factorizations to find the LCM (the first beat where both patterns land together).
  4. Verify by listing multiples of both numbers until they match.
  5. Find the GCF of 18 and 24 using the same prime factorizations.
18 24 3 (extra 3) 2 3 shared 2 2 (extra 2s) LCM = all unique primes at highest power = 2³ x 3² = ?

Step-by-Step Investigation Guide

  1. Build a factor tree for 18 Start by splitting 18 into any two factors (like 2 x 9 or 3 x 6). Keep splitting until every bottom number is prime. Circle the prime numbers.

    Does it matter which pair you start with? Will you get different prime factors?

  2. Build a factor tree for 24 Do the same for 24. Try starting with a different pair than another team member (one person tries 4 x 6, another tries 2 x 12). Compare results.

    Did different starting points give the same prime factorization? Why does that happen?

  3. Write each as a product of primes with exponents 18 = 2 x 3 x 3 = 2 x 3². Write 24 the same way. Using exponents makes it easier to compare the structures.

    Which prime appears more times in 24 than in 18?

  4. Find the LCM using prime factorizations Take each prime that appears in EITHER factorization. Use the HIGHEST power of that prime. Multiply them together.

    Why do we use the highest power, not the lowest?

  5. Verify by listing multiples Write multiples of 18: 18, 36, 54, 72... Write multiples of 24: 24, 48, 72... The first number on BOTH lists is the LCM. Does it match your answer from step 4?

    How does the list method confirm (or correct) your factor-tree method?

  6. Find the GCF using the same factorizations For the GCF, take each prime that appears in BOTH factorizations and use the LOWEST power. Multiply. Check: does LCM x GCF = 18 x 24?

    Why does the relationship LCM x GCF = product of the two numbers always work?

Language Support

Key Vocabulary

Prime number: A number with exactly 2 factors: 1 and itself (like 2, 3, 5, 7)
Composite number: A number with more than 2 factors (like 4, 6, 9, 12)
Factor tree: A diagram that breaks a number into smaller and smaller factors until all are prime
Prime factorization: Writing a number as a product of only prime numbers
LCM: Least Common Multiple - the smallest number that both numbers divide into
Exponent: The small number that shows how many times to multiply a base (3² means 3 x 3)

Sentence Frames

  • "The prime factorization of ___ is ___ because those are the only prime numbers that multiply to ___."
  • "I split ___ into ___ and ___ because ___ x ___ = ___."
  • "The LCM is ___ because it uses the highest power of each prime: ___."
  • "I know ___ is prime because its only factors are 1 and ___."

Multiple Representations

Factor Tree

Branch diagram splitting a number down to its prime factors at the leaves.

Venn Diagram

Place shared primes in the overlap. Unique primes go in each circle. LCM = all, GCF = overlap.

Listing Multiples

Write multiples of both numbers. The first match is the LCM.

Work Space

Factor Tree for 18:



Factor Tree for 24:



Prime Factorizations:


Venn Diagram:



LCM and GCF:


Defense: