Unit 1 Lesson 2 Practice Problems IM® Algebra 2TM authored by Illustrative Mathematics®

Britta Dwyer

21 Nov 202214:28

Summary

TLDRThe video script is an educational tutorial on geometric sequences, growth factors, and patterns in sequences. It explains how to find subsequent terms in a geometric sequence by multiplying by a constant growth factor. It also demonstrates calculating growth factors by dividing successive terms and applies this concept to a credit card debt example with compounding monthly interest. The script further explores the Sierpinski triangle, illustrating the pattern of shaded triangles and the relationship between the number of triangles and their area over iterations. Additionally, it discusses graphing these patterns and creating custom sequences based on given rules, offering insights into mathematical sequences and their applications.

Takeaways

🔢 In a geometric sequence, each term is found by multiplying the previous term by a constant factor, known as the growth factor.
📈 The growth factor can be determined by dividing a term by its preceding term in the sequence.
💳 A credit card balance with a 2% monthly interest rate increases by 102% of the previous month's balance, calculated by multiplying the balance by 1.02 each month.
📊 The number of shaded triangles in a Sierpinski triangle increases exponentially with each step, while the area of each triangle decreases.
📉 The graphs of the number of shaded triangles and the area of each triangle in a Sierpinski triangle demonstrate different growth patterns; the number of triangles increases while the area per triangle decreases.
🔑 The rule 'four less than three times the previous number' is used to generate sequences where each term is calculated by multiplying the previous term by three and then subtracting four.
📐 The Sierpinski triangle is constructed by recursively removing the middle triangle from a set of four congruent triangles, creating a fractal pattern.
📘 The area of each triangle in the Sierpinski triangle sequence is calculated by dividing the total area by the number of triangles at each step.
💡 The script demonstrates how to apply geometric sequence concepts to real-world scenarios, such as calculating compound interest on a credit card.
📌 The script illustrates the process of creating and analyzing sequences using different rules, showcasing the versatility of geometric sequences.

Q & A

What is the definition of a geometric sequence?
-A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
How do you find the next three terms of a geometric sequence if the first two terms are given?
-To find the next three terms of a geometric sequence, you multiply each term by the common ratio to get the subsequent term. For example, if the first term is 1 and the second term is 4, and the common ratio is 2 (since 1*2=2 and 2*2=4), the next three terms would be 4*2=8, 8*2=16, and 16*2=32.
What is the growth factor of a geometric sequence and how do you calculate it?
-The growth factor, also known as the common ratio, is the number by which you multiply to get from one term in a geometric sequence to the next. It can be calculated by dividing any term by its preceding term.
How does the growth factor affect the terms in a geometric sequence?
-The growth factor determines the rate at which the terms in a geometric sequence increase or decrease. A growth factor greater than 1 will increase the terms, while a growth factor less than 1 will decrease them.
If a credit card balance of $1000 has a 2% monthly interest rate and no payments are made, how does the balance change over time?
-With a 2% monthly interest rate, the balance increases by 2% each month. This means the balance is multiplied by 1.02 each month. For example, after one month, the balance would be $1000 * 1.02 = $1020.
What is the pattern of shaded triangles in the Sierpinski Triangle, and how does it relate to the number of steps taken?
-In the Sierpinski Triangle, at each step, the number of shaded triangles increases by a factor of three, and the area of each triangle is divided by four. This pattern continues with each iteration.
How can you graph the number of shaded triangles and the area of each triangle in the Sierpinski Triangle as a function of the step number?
-You can graph the number of shaded triangles by plotting the count on the y-axis against the step number on the x-axis, showing an exponential increase. For the area of each triangle, plot the area on the y-axis against the step number, showing an exponential decrease.
What is the rule for creating a sequence where each number is four less than three times the previous number?
-To create a sequence where each number is four less than three times the previous number, you multiply the previous number by three and then subtract four to get the next number in the sequence.
How does changing the starting number in a sequence affect the subsequent terms when the rule is 'four less than three times the previous number'?
-Changing the starting number in the sequence will result in a different set of numbers following the same rule. Each subsequent term will still be calculated by multiplying the previous term by three and subtracting four, but the values will be different based on the initial number.
Can you provide an example of a different rule for generating a sequence, and what would the next three terms be if starting with 1?
-An example of a different rule could be 'multiply by -1'. Starting with 1, the next three terms would be: 1 * -1 = -1, -1 * -1 = 1, and 1 * -1 = -1.

Outlines

00:00

🔢 Understanding Geometric Sequences

This paragraph explains the concept of geometric sequences through examples. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The paragraph illustrates this with the sequence starting with 1 and a common ratio of 2, resulting in terms 1, 2, 4, 8, 16, and 32. It also discusses how to find the growth factor of a sequence, which is the multiplier used to get from one term to the next, using examples including a sequence decreasing from 256 to 128 with a growth factor of 1/2, and another increasing from 0.08 to 0.8 with a growth factor of 10. Additionally, it covers a practical example of a credit card balance increasing by 2% per month, showing how the balance grows using the growth factor of 1.02.

05:03

📊 Geometric Sequences in Geometry: The Cinsky Triangle

The second paragraph delves into a geometric application of sequences, specifically the Cinsky triangle problem. It starts with an equilateral triangle and iteratively breaks it into four smaller congruent triangles, removing the middle one. The process is repeated for each remaining triangle. The paragraph aims to complete a table showing the number of shaded triangles and their respective areas after each step. The number of shaded triangles increases by a factor of three with each step, while the area of each triangle decreases as the total area is divided among more triangles. This results in a pattern where the number of triangles grows exponentially, but the area per triangle decreases. The paragraph also includes instructions for graphing these sequences separately, showing how the number of shaded triangles increases while the area per triangle decreases with each step.

10:07

📐 Creating Custom Sequences with Mathematical Rules

The third paragraph introduces the creation of custom number sequences using specific rules. It presents a rule where each number is four less than three times the previous number, starting with 10, and builds a sequence of five numbers following this rule. The paragraph also encourages the selection of different starting numbers and the creation of sequences based on various rules, such as multiplying by -1 or subtracting two, to demonstrate the flexibility in sequence generation. This section showcases how different mathematical operations can be applied to create diverse sequences, emphasizing the importance of understanding the operations involved in sequence construction.

Mindmap

Keywords

💡Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the video, the concept is introduced with an example where each term is multiplied by two to find the subsequent term, illustrating the core idea of geometric progression.

💡Growth Factor

The growth factor refers to the common ratio in a geometric sequence, which determines the rate at which the sequence grows. In the script, the growth factor is calculated by dividing a term by its preceding term, such as determining the growth factor of a sequence progressing from 256 to 128 by calculating 128/256, which simplifies to 1/2.

💡Interest Rate

The interest rate mentioned in the script is the percentage charged on a loan, credit, or deposit per period of time. In the context of the video, it's used to calculate the monthly balance on a credit card with a 2% monthly interest rate, showing how the balance increases by 2% each month if no payments are made.

💡Cinski Triangle

The Cinski Triangle, or Sierpinski Triangle, is a fractal and geometric pattern that forms a specific shape when an equilateral triangle is recursively subdivided. In the video, the process is described where each iteration removes the middle of four smaller triangles, creating a pattern that is used to discuss the number of shaded triangles and their areas.

💡Area

Area, in the context of the video, refers to the amount of space enclosed within a two-dimensional shape. The script discusses how the area of the triangles in the Sierpinski Triangle pattern changes with each iteration, decreasing as the number of triangles increases, which is a key aspect of understanding the geometric properties of the pattern.

💡Shaded Triangles

Shaded triangles in the video script refer to the individual triangles that are filled in during the construction of the Sierpinski Triangle. The number of shaded triangles increases with each step of the process, which is a key aspect of the pattern's progression and is used to discuss the growth of the pattern.

💡Graph

A graph in the video is used as a visual representation of data, showing the relationship between two variables. The script describes how to graph the number of shaded triangles and the area of each triangle as functions of the step number in the Sierpinski Triangle construction, providing a visual tool for understanding the changes over time.

💡Sequence

A sequence in the video refers to an ordered list of numbers or objects. The script discusses creating sequences based on specific rules, such as multiplying by a certain factor and then subtracting a number, which is a method for generating a series of numbers that follow a predictable pattern.

💡Rule

In the context of the video, a rule is a mathematical operation or set of operations that dictate how a sequence progresses. For example, the rule 'four less than three times the previous number' is used to generate a sequence by first multiplying the previous term by three and then subtracting four, showcasing how rules define the behavior of sequences.

💡Decimal Place

The decimal place in the video refers to the position of digits in a decimal number, which can affect the value of the number. The script mentions a sequence where the decimal place increases by one with each term, demonstrating how the position of digits can influence the growth of numbers in a sequence.

Highlights

Geometric sequence is defined by a constant multiplication factor between terms.

The next three terms of a sequence are found by multiplying each term by the growth factor.

The growth factor for a sequence where each term is doubled is two.

To find the growth factor, divide the new term by the original term.

A sequence with a growth factor of one remains constant.

A sequence with a growth factor of 1/2 decreases by half with each term.

A sequence with a growth factor of 1/10 decreases by a factor of ten with each term.

A credit card balance with a 2% monthly interest rate increases by 102% of the previous month's balance.

The growth factor for a 2% monthly interest rate is 1.02.

The Sierpinski triangle is formed by repeatedly removing the middle triangle from a set of four.

The number of shaded triangles in the Sierpinski triangle pattern doubles with each step.

The area of each triangle in the Sierpinski pattern is one-fourth of the previous step's area.

The growth pattern of the Sierpinski triangle can be graphed to show the increasing number of triangles and decreasing area.

The number of shaded triangles in the Sierpinski triangle pattern increases exponentially with each step.

The area of each triangle in the Sierpinski pattern decreases as more triangles are formed.

A sequence where each number is four less than three times the previous number starts with a given initial value.

Different starting numbers yield different sequences following the same rule.

Alternative rules for sequence generation can be multiplication by a constant or subtraction by a constant.