Square Roots and Cube Roots Using Prime Factorization

Outcome: FP10.1 – Demonstrate understanding of factors of whole numbers by determining the:

• prime factors
• greatest common factor
• least common multiple
• principal square root
• cube root.

Learning Goals:

1. Any prime number that divides another number evenly is a prime factor of that number.
2. If the prime factors of a number can be evenly divided into two groups with the same numbers in each group, the number is a perfect square, and its square root is the factors in one group multiplied together.
3. If the prime factors of a number can be evenly divided into three groups with the same numbers in each group, the number is a perfect cube, and its cube root is the factors in one group multiplied together.
4. Factors of a number can be grouped together – two factors of 2 = 2²

Task/Problem/Question: There are two trees in the forest, one’s age is a perfect square, and the other is a perfect cube. Both have 12 prime factors. What are the possible ages of the two trees?

Alternative: There are two trees in the forest, one’s age is a perfect square, and the other is a perfect cube. What are their possible prime factorizations?

Anticipation:

1. What do you think students might do?
1. Student writes all solutions as one repeated prime number. (2*2*2*2*2*2*2*2*2*2*2*2). Questions to Ask: How can those factors be grouped? Can they be written in any other way? How would you find an answer with more than one prime number?
2. Solutions are found by choosing smaller numbers and adding 1’s as needed. (4= 2*2*1*1*1*1*1*1*1*1*1*1) Questions to Ask: Why does adding 1 work? How can you find an answer without 1 being a prime factor that is written? Is 1 really a prime factor, anyway?
3. Guess and check method, pick a number, see if it is a perfect square/cube, see if it has twelve factors. Questions to Ask: Can you see any patterns in the factors of perfect squares and perfect cubes? Does the pattern persist through all perfect squares and cubes? How can that pattern help you find a perfect square or cube with twelve prime factors?
4. Try out different combinations of prime factors to see if they result in a perfect cube/square. Questions to Ask: What types of prime numbers are you using? How are they being chosen? What methods could you try to come up with prime numbers to use?
5. Recognize that the solution must have an equal number of each prime factor in either two groups or three (perfect square or cube), start balancing the two or three groups. Questions to Ask: How do you know that there needs to be two/three groups?

1. What barriers might the students encounter?
1. Using non-prime numbers as factors. Questions to Ask: Is that the smallest that particular factor can be broken down? Is **** divisible by any other numbers? What is the difference between a prime number and a composite number?
2. Using the wrong number of factors. Questions to Ask: Can you make a perfect square with an odd number of prime factors? What if there were more/less factors in the trees age, like twelve? How would that change your answers?
3. Not having understanding of what a perfect square, a perfect cube or a prime factor is. Questions to Ask: What do numbers like 4, 25, and 36 have in common? What are the side lengths on a square? What are the length, width and height on a cube? Is that the smallest that particular factor can be broken down? Is **** divisible by any other numbers? What is the difference between a prime number and a composite number?
4. Not knowing where to start, not understanding what the question wants. Questions to Ask: What are some numbers you can begin working with? (lead students to either beginning with factors or the whole numbers) What are some perfect squares or cubes that you already know? How do you do the prime factorization of these numbers?

Monitoring:

 Repeated Primes Adding 1 as needed Guess and check with perfect cubes and squares Guess and check with prime factors Balancing groups of prime factors Other

Selecting and Sequencing:

1. Move from guess and check methods into methods that recognize patterns.
2. Start with groups looking at the whole number first and move to students who looked at the prime factors first and vice versa.
3. Move from repetition of the same number to adding ones as needed to using many different prime numbers.

Connecting:

What are you going to say or ask the students to get them to connect their work to the learning goals?

1. How many of each prime number do we need to make factors of perfect squares? Perfect cubes? Alternate Wording: Can we have one 5 be a factor of a perfect square or cube? Why or why not?
2. How can we write the factors to group them better?
3. How can the square/cube root of the number be represented by the prime factors?
4. Why does adding 1 as needed to the prime factors work?
5. What are some ways of showing all of the numbers that have twelve factors and are perfect squares or cubes?
6. What are some numbers that are both perfect squares and cubes that have twelve factors? Can this group be represented algebraically?
7. Can two prime factors that are the same be combined in different ways? Would combining them in any of these ways be useful?
8. Using the rules we have created for square and cube roots, could we also expand these rules for other roots? Like a fourth root or a sixth root? Could we determine if an object has any roots using these methods?
9. Is 1 a prime number? Could we use the answers that include multiple 1s as examples of numbers with 12 prime factors?