Problems to Ponder A focus on Strategies, Collaboration, and Explanation of Solutions Math Teacher Circle May 18, 2017.

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Problems to Ponder A focus on Strategies, Collaboration, and Explanation of Solutions Math Teacher Circle May 18, 2017

Ground Rules First take a few minutes to think about the problem and attempt a solution on your own. (THINK) Second, work with your table partners to share initial strategies and decide which one(s) offer the best opportunity to reach a solution (PAIR/GROUP) Finally, appoint a spokesperson from your group to share your group’s solution(s) when invited to participate in the whole group discussion (SHARE)

Locker Problem Problem: If, in a school of 500 lockers, one student opens every locker, a second student, beginning at the second locker, closes every second locker, a third student, beginning at locker three, changes every third locker and so on until the 500th student changes the 500th locker, which lockers are then open? Explain your reasoning fully. Extensions: What if only the even numbered students change lockers? What if only odd numbered students? What if prime numbered students? What if Fibonacci numbered students?

Locker Solutions 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Locker Solutions Those with an odd number of factors will be open, i.e. numbers of the form 𝑛2 Extensions: What if only the even numbered students? The open lockers are those of the form 2𝑛2 What if only odd numbered students? The open lockers are those of the form 2 𝑚 𝑛 2 What if only prime numbered students? What if only Fibonacci numbered students?

Sum of Consecutive Natural Numbers Problem: Find the sum of the first 100 natural numbers. Extension: Find the sum of the first 1000 natural numbers. Extension: Find the sum of the first 𝑛 natural numbers. Strategies? Visual representations?

Staircase Problem Problem: How many blocks would it take to construct a staircase whose tallest stair is made from 𝑛 cubes?

Staircase Solution Example: 5 cubes in the tallest stair

Staircase Solution 5×6 2 =15 blocks Example: 5 cubes in the tallest stair 5 6 5×6 2 =15 blocks

Staircase Solution # blocks= 𝑛× 𝑛+1 2 General Solution: Number of blocks in a staircase with tallest stair of height 𝑛 𝑛 𝑛+1 # blocks= 𝑛× 𝑛+1 2

Number Sequence Problems Problem: Find the next number in the following sequence: 5, 11, 19, 29, 41, 55, … Extensions: What number comes after that? How could you describe the pattern? What would be the 20th number in this sequence? What about the 100th number? What is the value of the 𝑛th term in this sequence?

Visual Representation of Sequence We have the sequence: 5, 11, 19, 29, 41, 55, … What is the 100th term in this sequence? What is the value of the 𝑛th term in this sequence?

Sequence Solutions We have the sequence: 5, 11, 19, 29, 41, 55, … What is the 20th term in this sequence? 461 What is the 100th term in this sequence? 10,301 What is the value of the 𝑛th term in this sequence? 𝑎𝑛 = (𝑛+1)2 + 𝑛 or 𝑎𝑛 = 𝑛2 + 3𝑛 + 1

Another Number Sequence Problem: Find the pattern in the following sequence: 7, 11, 15, 19, 23, 27, … Extensions: What would be the 100th number in this sequence? What is the value of the 𝑛th term in this sequence? Visual representation?

Sequence Solutions We have the sequence: 7, 11, 15, 19, 23, 27, … What would be the 100th number in this sequence? 403 What is the value of the 𝑛th term in this sequence? 𝑎𝑛 = 2 2𝑛 + 1 +1 or 𝑎𝑛 = 4𝑛 + 3

Another Number Sequence Problem: Find the pattern in the following sequence: 4, 8, 14, 22, 32, 44, … Extensions: What would be the 100th number in this sequence? What is the value of the 𝑛th term in this sequence? Visual representation?

Sequence Solutions We have the sequence: 4, 8, 14, 22, 32, 44, … What would be the 100th number in this sequence? 10,102 What is the value of the 𝑛th term in this sequence? 𝑎𝑛 = 𝑛(𝑛 + 1) + 2 or 𝑎𝑛 = 𝑛2 + 𝑛 + 2

Start with a Visual Representation Problem: What is this sequence? Extensions: What would be the 100th number in this sequence? What is the value of the 𝑛th term in this sequence?

Visual Representation Solutions Problem: What is the sequence? 4, 8, 12, 16, 20, … Extensions: What would be the 100th number in this sequence? 400 What is the value of the 𝑛th term in this sequence? 𝑎𝑛 = (𝑛 + 1)2 – (𝑛 − 1)2 or 𝑎𝑛 = 4𝑛

Factor Fascination Problem: Which natural numbers have exactly two factors? Which have exactly three factors? Which have exactly four factors? Which have exactly six factors? What can you generalize about each of these categories of numbers? Extension: Continue your exploration for natural numbers with five, seven, eight, and nine factors. Explain your discoveries so far.