Illustrative mathematics algebra 2 unit 1 answer key

Problem 1

Find the sum of the sequence:

1. \(\frac 1 3 + \frac 1 9\)
2. \(\frac 2 3 + \frac 2 9\)
3. \(\frac 1 3 + \frac 1 9 + \frac{1}{27}\)
4. \(\frac 2 3 + \frac 2 9 + \frac{2}{27}\)
5. \(\frac 1 3 + \frac 1 9 + \frac{1}{27} + \frac{1}{81}\)
6. \(\frac 2 3 + \frac 2 9 + \frac{2}{27} + \frac{2}{81}\)

Problem 2

Priya is walking down a long hallway. She walks halfway and stops. Then, she walks half of the remaining distance, and stops again. She continues to stop every time she goes half of the remaining distance.

1. What fraction of the length of the hallway will Priya have covered after she starts and stops two times?
2. What fraction of the length of the hallway will Priya have covered after she starts and stops four times?
3. Will Priya ever reach the end of the hallway, repeatedly starting and stopping at half the remaining distance? Explain your thinking.

Problem 3

A geometric sequence \(h\) starts with 10, 5, . . . Explain how you would calculate the value of the 100th term.

Problem 4

Here is a graph of sequence \(r\). Define \(r\) recursively using function notation.

Problem 5

An unfolded piece of paper is 0.05 mm thick.

1. Complete the table with the thickness of the piece of paper \(T(n)\) after it is folded in half \(n\) times.
2. Define \(T\) for the \(n^{\text{th}}\) term.
3. What is a reasonable domain for the function \(T\)? Explain how you know.

Problem 6

A piece of paper is 0.05 mm thick.

1. Complete the table with the thickness of the paper \(t(n)\), in mm, after it has been folded \(n\) times.
2. Does \(t(0.5)\) make sense? Explain how you know.

Problem 7

An arithmetic sequence \(a\) starts 84, 77, . . .

1. Define \(a\) recursively.
2. Define \(a\) for the \(n^{\text{th}}\) term.

Problem 8

Here is a pattern of growing rectangles:

1. Describe how the rectangle grows from Step 0 to Step 2.
2. Write an equation for sequence \(S\), so that \(S(n)\) is the number of squares in Step \(n\).
3. Is \(S\) a geometric sequence, an arithmetic sequence, or neither? Explain how you know.

Lesson 1

A Towering Sequence

Lesson Narrative

The purpose of this lesson is for students to work with sequences and describe them recursively in an informal way. A sequence is defined here as a list of numbers while a term (of a sequence) is one of the numbers in the list.

Using the Tower of Hanoi puzzle, students first make sense of the rules of the puzzle before playing with different numbers of discs in order to generate a sequence representing the minimum number of moves needed to complete the puzzle (MP1). Solving the puzzles provides students opportunities to express regularity in repeated reasoning (MP8) when they informally state a recursive rule for generating the next term in a sequence from the previous term. If students try to convince each other that they have found the minimum number of moves to solve the puzzle, they are constructing mathematical arguments and critiquing the reasoning of others (MP3). While geometric and arithmetic sequences are a focus of the unit, the sequence generated from the Tower of Hanoi is neither. This was a deliberate choice to promote student discussion since the pattern to generate the sequence is one students are less likely to guess correctly than a more straightforward linear or exponential pattern.

The second activity is optional and provided for additional practice with another context if needed, or as an alternative activity if students have familiarity with the Tower of Hanoi puzzle.

Learning Goals

Teacher Facing

• Comprehend the term "sequence" (in written and spoken language) as a list of numbers.

• Describe (orally) a recursive rule for identifying the next term of a simple sequence.

• Generate a sequence that arises from a mathematical context.

Student Facing

• Let’s explore the Tower of Hanoi.

Required Materials

Required Preparation

Students should manipulate either physical or digital objects to experiment with the Tower of Hanoi puzzle. For physical, each group could use a quarter, nickel, penny, and dime, and a piece of paper with 3 circles drawn on it. For students using the digital version of the materials, acquire devices that can run the digital applet. It is ideal if each student has their own device.

If you are doing the optional checker jumping puzzle, each group needs at least 3 tokens each of 2 different colors. These could be actual checkers, counting chips, pennies and nickels, or any other appropriate tokens.

Learning Targets

Student Facing

• I can give an example of a sequence.

CCSS Standards

Glossary Entries

• sequence

  A list of numbers, possibly going on forever, such as all the odd positive integers arranged in order: 1, 3, 5, 7, . . . .

• term (of a sequence)

  One of the numbers in a sequence.

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