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You did these with your group. Now do them on your own — the model above is yours to copy.

1. 1

   Same stepsFinish what the group started. Use the model above, step for step.

   Try it together — then prove it: Find the prime factorization of 12. Then explain what happens if you start a different way. Start with 12 = 3 × 4. Split until every branch is prime. (4 = 2 × 2, so 12 = 2 × 2 × 3) Write the answer with an exponent. (2² × 3) Ty starts with 12 = 2 × 6. Will he get different primes? Why? (No. 6 = 2 × 3, so he also gets 2 × 2 × 3. The primes are always the same.)

   Show your work
2. 2

   Warm restartWhat is the LCM of 6 and 9?

   1. A3
   2. B18
   3. C27
   4. D54

   How do you know?
3. 3

   Warm restartList the first four multiples of 7.

   1. A1, 7, 14, 21
   2. B7, 14, 21, 28
   3. C7, 7, 7, 7
   4. D7, 17, 27, 37

   How do you know?
4. 4

   Warm restartWhat is the LCM of 5 and 10?

   1. A5
   2. B10
   3. C15
   4. D50

   How do you know?

You answered these out loud. Now write them.

Answer, then say how you know — the reason is the part that counts.

1. 5

   Think it throughWhat is the prime factorization of 72?

   1. A2³ × 3²
   2. B2² × 3³
   3. C6 × 12
   4. D8 × 9

   Why is that the answer?
2. 6

   Think it throughThe gardener wants a grid with 8 rows. How many pods in each row so all 72 are used?

   1. A6
   2. B9
   3. C12
   4. D64

   How do you know? Give a second reason as well.
3. 7

   Think it throughWhich grid would NOT use exactly all 72 seed pods?

   1. A3 by 24
   2. B4 by 18
   3. C5 by 14
   4. D6 by 12

   Explain your thinking. What would have to change for a different choice to be right?
4. 8

   Back to the modelHow did you decide whether a number is prime or composite? What strategy did you use?

Use the words

Fill each blank with a word from the bank.

• Writing a number as ___ numbers multiplied together is its prime factorization.
• A whole number greater than 1 with exactly two factors, 1 and itself, is a ___ number.
• A whole number greater than 1 that has more than two factors is a ___ number.
• A number that divides evenly into another number is called a ___.

Explain the thinking

1. 9

   Find the mistakeAnother student made this exact mistake in this lesson. Explain why it is wrong, then write what they should have done.

   The mistake: A common mistake in Prime Factorization is stopping the factor tree too soon — leaving a composite number like 4, 6, 8, or 9 in the final answer instead of continuing to split it.
2. 10

   Say moreLook at your factor tree for 60. How did you decide which numbers to break apart first, and how do you know when to stop?
3. 11

   Write your ownWrite your own problem about this situation, then solve it and show the answer. The situation: Engineers on Station Helios need to break 84 replacement circuit boards into equal shipping crates, and the crates must be built using only prime-numbered groupings.

   Show your work

Last check

These two come from Lesson 6.12. If they are shaky, that is the lesson to revisit — not this one.

1. 12

   Show what you knowExplain your thinking — What is the prime factorization of 40?

   1. A2 × 2 × 2 × 5
   2. B2 × 20
   3. C4 × 10
   4. D5 × 8

   Explain your choice.
2. 13

   From Lesson 6.12Explain your thinking — What is the LCM of 9 and 12?

   1. A3
   2. B36
   3. C72
   4. D108

   How do you know?
3. 14

   From Lesson 6.12How many packs of buns do you buy?

   1. A3
   2. B4
   3. C5
   4. D6

   How do you know?

Be honest — this tells your teacher what to do next