<p>Students will study notations that found in patterns and sequences in this unit. Students are going to:<br>- learn about the Fibonacci Sequence, the&nbsp;Golden Ratio, Pascal's Triangle, and its connections.<br>- look into geometric numbers and patterns, and make connections between geometric numbers.<br>- examine convergence and divergence concepts and sequences.</p>

<p>- What notations are widely accepted in mathematics? How do notations contribute to mathematical understanding?<br>- How does the process of solving problems involve the usage of mathematical notations?<br>&nbsp;</p>

<p>- <strong>Arithmetic Sequence:</strong> A sequence, whereby the terms increase by a constant amount.&nbsp;<br>- <strong>Converge:</strong> To approach one specific number.&nbsp;<br>- <strong>Diverge:</strong> Not converging; for a series, one that has no bounded sum.&nbsp;<br>- <strong>Fibonacci Sequence:</strong> A sequence obtained by adding the two previous terms to find the next term.&nbsp;<br>- <strong>Geometric Sequence:</strong> A sequence, whereby the terms increase by a constant multiplier, <i>r</i>.&nbsp;<br>- <strong>Geometric Series:</strong> The indicated sum of a finite or ordered infinite set of terms. It is infinite or finite according as the number of terms is infinite or finite. A geometric series is one whose terms form a geometric progression. The general term of a geometric series is <i>a</i> + <i>ar</i> + <i>a\(r^2\)</i> + <i>a\(r^3\)</i> + … + <i>a\(r^{n-1}\)</i> + …Its sum to n terms is \(S_n\) = \(a(1 - r^n) \over 1 - r\).&nbsp;<br>- <strong>Golden Ratio:</strong> The “extreme and mean ratio,” coined by Euclid.&nbsp;<br>- <strong>Pascal’s Triangle:</strong> A triangular formation of patterns.&nbsp;<br>- <strong>Pattern:</strong> A repetition of some sort.&nbsp;<br>- <strong>Recursive:</strong> For a function, one whose implementation references itself.&nbsp;<br>- <strong>Sequence:</strong> A list of numbers, symbols, or objects that either follows or does not follow a pattern.&nbsp;<br>- <strong>Square Number:</strong> A number that creates a square, the second power of an integer, i.e., 1, 4, 9, 16,….</p>

<p>- copies of Arithmetic Sequences (M-A1-2-3_Arithmetic Sequences)<br>- copies of the Squares worksheet (M-A1-2-3_Squares and KEY)<br>- copies of the Fractals handout (M-A1-2-3_Fractals)<br>- Pascal’s Triangle handout (M-A1-2-3_Pascal Triangle)</p>

<p>- Examine and observe the class discussion. Each student's contribution fulfills a quality requirement if it is original, specific to the mathematical material, and contributes to the class's collective knowledge. Student questions and answers can also progress the conversations.&nbsp;<br>- Analyze students' work on their investigations into different number types and the relationship between square and triangle numbers (Are the computations and patterns accurate?).&nbsp;<br>- An illustration of the Fibonacci Sequence's Golden Ratio.&nbsp;<br>- An article discussing the usage and prominence of the Golden Ratio (Is the 1.6 approximation appropriate?).&nbsp;<br>examination of the patterns in Pascal's Triangle.&nbsp;<br>- Connection between Pascal's Triangle and the Fibonacci Sequence.&nbsp;<br>illustration of geometric and arithmetic sequences (Did any students notice that some mathematical patterns connect subjects that don't seem to go together, like algebra and geometry?).&nbsp;<br>&nbsp;</p>

<p>Explicit Instruction, Modeling, Scaffolding, and Active Engagement&nbsp;<br>W: Students look into geometric numbers, such as square and triangle numbers and their relationships, as well as particular patterns and connections found in relationships between various representations and convergence/divergence in sequences. This work presents a broad perspective on pattern and sequence learning.&nbsp;<br>H: Student conversation is facilitated by the study of geometric number diagrams that are visually presented and by using different methods to look for patterns.&nbsp;<br>E: The lesson is divided into two parts. Part 1 concentrates on different kinds of numbers and patterns, while Part 2 discusses patterns and sequences.&nbsp;<br>R: Students are required to reflect, revisit, rewrite, and reconsider throughout the lesson due to the significant work of identifying patterns, rules, and linkages between notations/representations.&nbsp;<br>E: Throughout the course, students are required to conduct self-evaluations, particularly when drawing links between Pascal's Triangle, the Golden Ratio, and the Fibonacci Sequence. Regarding the self-evaluation that students are required to complete, at least one task needs to be modeled. Leading questions and techniques for contrasting students'&nbsp;work with one or more predicted outcomes should be incorporated into the modeling exercise.&nbsp;<br>T: Several learning modalities are targeted by the incorporation of written and spoken activities. When necessary, support is provided through the implementation of both individual and group activities.&nbsp;<br>O: The lesson is quite abstract in nature. students are required to identify patterns and make conjectures.</p>

<p><strong>Part 1: Types of Numbers and Patterns: Geometric Numbers, Including Triangular and Square Numbers</strong><br><br><strong>First Five Square Numbers</strong></p><figure class="image"><img style="aspect-ratio:435/117;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="435" height="117"></figure><p><strong>"Which numbers do the displayed diagrams represent? What kind of numbers are these?"</strong> Students should understand the relationship between the shape of the squares and the concept of <i>square numbers</i> (i.e., square numbers form a square).<br><br><strong>"The square numbers we currently have are 1, 4, 9, 16, and 25."</strong><br><br>Draw four more square numbers. <strong>"Let's look at the pattern and try to break it into pieces. Remember, a </strong><i><strong>pattern</strong></i><strong> is simply a repetition of some sort. Suppose we have this table:&nbsp;</strong></p><figure class="image"><img style="aspect-ratio:250/596;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_48.png" width="250" height="596"></figure><p><strong>Which pattern are we creating? Is there any method to analyze this pattern? How do the expressions relate back to the diagrams? Can you see the summations and increasing patterns in the diagrams?"</strong><br><br><strong>"How about we look at another table in the form?</strong></p><figure class="image"><img style="aspect-ratio:248/574;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_49.png" width="248" height="574"></figure><p><strong>Does this table help you find a pattern? Assume we add the following columns:</strong></p><figure class="image"><img style="aspect-ratio:517/574;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_50.png" width="517" height="574"></figure><p><strong>Fill up the table. What have we discovered? What kind of pattern results? What does the knowledge of differences do for us?" </strong>Distribute the Squares worksheet (M-A1-2-3_Squares with KEY). Students should be encouraged to realize that the second difference is constant.&nbsp;<br><br><strong>"Develop a rule to determine the </strong><i><strong>n</strong></i><strong>th term of the square number sequence."</strong> (<i>a (n) = \(n^2\)</i>)&nbsp;<br><br><i>Important:</i> Students should understand that they do not need to use the difference columns to determine the rule. However, the columns do provide a foundation for understanding how the number of differences is related to the type of function.<strong> "The equation 𝒂(𝒏)= can be used to get the </strong><i><strong>n</strong></i><strong>th term. The fact that the second difference is constant and hence linear indicates that the function is a parabola (i.e., will have a highest power of 2).</strong><br><br><strong>"The concept of square numbers also suggest various geometric configurations of numbers. Triangle, pentagonal, and hexagonal numbers operate similarly to square numbers. Consider these representations."</strong><br><br>Triangular Numbers</p><figure class="image"><img style="aspect-ratio:435/117;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="435" height="117"></figure><p>Pentagonal Numbers</p><figure class="image"><img style="aspect-ratio:460/122;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_52.png" width="460" height="122"></figure><p>Hexagonal Numbers</p><figure class="image"><img style="aspect-ratio:255/352;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_53.png" width="255" height="352"></figure><p><strong>Activity 1: Enrichment</strong><br><br>Divide the students into groups of three or four. Each group should complete the following table:</p><figure class="image"><img style="aspect-ratio:597/376;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_54.png" width="597" height="376"></figure><p>Guide students through the concept of "difference of differences." For instance, with the square numbers, the second difference is linear (that is, it increases by 2 for each additional term).<br><br>Tell students, <strong>"Use any representation you'd like to recognize patterns and attempt to identify rules."</strong> Note:<br><br>The rule for the triangular numbers is: 𝒂(𝒏) = \(n(n + 1) \over 2\).<br><br>The rule for the pentagonal numbers is: <i>𝒂(𝒏)</i> = \(n(3n - 1) \over 2\) .<br><br>The rule for the hexagonal numbers is: <i>𝒂(𝒏)</i> = (<i>n</i>)(2<i>n </i>- 1) .<br><br>In order to demonstrate the accuracy of the rule, students will present at least one illustration and discuss their group's findings.<br><br><strong>Connection of Triangular Numbers and Square Numbers</strong><br><br><strong>"Have you ever considered the relationship between these types of numbers? What is the relationship between triangular numbers and square numbers? Is there a connection? How did you determine the relationship?"</strong><br><br><strong>Activity 2</strong><br><br>Divide the students into groups of three or four. Encourage students to investigate the connection between triangular and square numbers. Students must present a detailed explanation with at least two different diagrams provided for the connection. They may use tables, diagrams, or other forms of representation. A rule that relates the two types of numbers must also be specified. End the discussion by identifying and supporting reasons for the necessity and significance of such a relationship. The relationships between numbers in patterns might produce unexpected results. Refer to Related Resources for some resources to use.<br><br><strong>Part 2: Patterns and Sequences: Fibonacci Sequence, Golden Ratio, Pascal’s Triangle, and Connections Between Them</strong><br><br><strong>"Have you heard of the Fibonacci Sequence? First, we need to define </strong><i><strong>sequence</strong></i><strong>. What is a sequence? A </strong><i><strong>sequence</strong></i><strong> is a collection of numbers, symbols, or things that either follow a pattern or do not. The </strong><i><strong>Fibonacci Sequence</strong></i><strong> is formed by adding the two previous terms to determine the following term. Here's the sequence:</strong><br><br><strong>1, 1, 2, 3, 5, 8, 13,…</strong><br><br><strong>How might we recursively write this sequence? In other words, if we know the previous two terms, how can we create a rule to find the </strong><i><strong>n</strong></i><strong>th term?"</strong><br><br><strong>"The recursive rule for the Fibonacci Sequence is:</strong><br><br><strong>\(a_n\) = \(a_{n-1}\) + \(a_{n - 2}\), where </strong><i><strong>𝒏</strong></i><strong> ≥ 2.</strong><br><br><strong>So, why do we write? Notice that the first term has a value of one, and there are no preceding terms to add. However, if we start with </strong><i><strong>n</strong></i><strong> = 2, we can add 1 and nothing (or zero) to get 1, which is indeed the value of the second term."</strong><br><br><strong>"Why is this sequence essential? Does it occur outside of mathematics?" </strong>Lead students through a discussion of the Fibonacci sequence, which appears in science and nature.<br><br>&nbsp;</p><p><strong>The Golden Ratio</strong><br><br><strong>"While discussing the Golden Ratio, we shall address the following questions:</strong><br><br><strong>What is the Golden Ratio?</strong><br><strong>What is the Golden Ratio's value?</strong><br><strong>How may we represent the Golden Ratio?</strong><br><strong>How does the unit of measurement relate to the Fibonacci Sequence or to real-world objects?"</strong><br><br><strong>"Of course, the Golden Ratio is a ratio that appears in many different contexts. The true definition of the </strong><i><strong>Golden Ratio</strong></i><strong> was coined by Euclid as 'the extreme and mean ratio.'&nbsp;He used a line segment drawing to depict a proportion and determine the ratio that represented the Golden Ratio."</strong><br><br><strong>Assume that line segment </strong><i><strong>a</strong></i><strong> + </strong><i><strong>b</strong></i><strong> is available.</strong></p><figure class="image"><img style="aspect-ratio:149/78;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_55.png" width="149" height="78"></figure><p><strong>We can write the proportion:</strong><br><br><strong>\(a + b \over a\) = \(a \over b\)</strong><br><br><strong>With \(a \over b\)&nbsp;representing the Golden Ratio.”</strong><br><br><strong>"Using this proportion and the Golden Ratio, how can we verbally relate what we're saying to the illustration? In other words, what do the proportions and ratios indicate?"</strong><br><br><strong>"What else do we know about the golden ratio?</strong><br><strong>The value is&nbsp;1.6180339887…</strong><br><strong>The ratio is irrational and incommensurable.</strong><br><strong>The ratio can be calculated by comparing various quantities found all around us, including in science and nature."</strong><br><strong>“Based on the following notations, the Golden Ratio can be identified.”</strong></p><figure class="image"><img style="aspect-ratio:139/130;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_56.png" width="139" height="130"></figure><p><strong>"These notations are interchangeable. Notice how we can display the Golden Ratio using drawings, words, and symbols."</strong><br><br><strong>Activity 3</strong><br><br>Provide a thorough and understandable explanation and example to help students justify the following claim: As <i>n</i> grows larger and closer to infinity, the ratio of any two consecutive Fibonacci terms approaches the Golden Ratio.<br><br><strong>Activity 4</strong><br><br>Have students write a brief article in pairs about the prevalence of the Golden Ratio in the world around them. Select one specific example and demonstrate how that object/structure represents the Golden Ratio. It is necessary to have at least one drawing. Students are free to utilize tables, symbols, and whatever other representations they require. They must clearly define and illustrate how the Golden Ratio is demonstrated in a real world example.<br><br>Ask each pair to present their articles and artwork to the class. They can make a class collage depicting all of the numerous representations of the Golden Ratio. Illustrations may contain polygons, rectangular pyramids, seeds, petals, shells, other pyramids, and so on. Student writings should properly clarify the Golden Ratio used in the examples.<br><br><strong>Activity 5: Pascal’s Triangle</strong><br><br>Show students examples of fractal geometry by using resources like <a href="http://en.wikipedia.org/wiki/File:Von_Koch_curve.gif"><span style="color:#1155cc;"><u>http://en.wikipedia.org/wiki/File:Von_Koch_curve.gif</u></span></a>. This website displays an animation of the Koch snowflake. Distribute the Fractals handout, which includes Sierpinski's Triangle and four variants of the Koch snowflake (M-A1-2-3_Fractals).<br><br><strong>"Have you heard of Pascal's Triangle? How is Pascal's Triangle created? What patterns are displayed?" </strong>Allow students to draw a triangle. <strong>"Pascal's Triangle is a triangular formation of patterns."</strong><br><br>Divide the students into groups of three or four. Students should investigate the patterns observed inside Pascal's Triangle (M-A1-2-3_Pascal Triangle). They should respond to the following questions:<br><br>What number sets or patterns can you discover inside the triangle? (<i>For example, </i>the natural number set is found within the second diagonal.)<br>Can you find the triangular numbers inside the triangle? Any additional numbers?<br>How would you explain the sum of the numbers in each row of the triangle?<br>Lead students to discover that the sum of the numbers in each row is a power of 2. Create a table with headings for row number, total of numbers, and an alternative way to write the sum. After students have created the table, ask them to find the rule for the sum of the numbers in the <i>n</i>th row. (The sum for row <i>n</i> is \(2^{n-1}\). The pattern is&nbsp;<br><br>1 + 2 + 4 + 8 + 16 +...<br><br><strong>Note: </strong>Pascal's Triangle demonstrates that the total of the diagonals matches the number in the following row. Refer to the Pascal Triangle handout (M-A1-2-3_Pascal Triangle) to observe how the sums progress in subsequent rows. For example, adding 1, 4, 10, and 20 yields 35. This number may be found in the next row, below and to the left of the 20!<br><br><strong>Activity 6</strong><br><br>Divide students into groups of three or four. Each group should investigate how to connect Pascal's Triangle to the Fibonacci sequence. In other words, they can find the Fibonacci sequence inside Pascal's Triangle. They should particularly identify the Golden Ratio, which may be obtained from the triangle.<br><br><strong>Arithmetic Sequences</strong><br><br><strong>“Assume the following sequence is represented on a graph.”</strong></p><figure class="image"><img style="aspect-ratio:315/352;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_57.png" width="315" height="352"></figure><p><strong>"Let's write this sequence: -9, -6, -3, 0, 3, 6, ... on the board. Then, let's write it in a table:</strong></p><figure class="image"><img style="aspect-ratio:236/636;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_58.png" width="236" height="636"></figure><p><strong>What is happening here? How can we characterize what occurs from term to term? Can we predict the next term based on the preceding term? Can we find the </strong><i><strong>n</strong></i><strong>th phrase without knowing the preceding one? If so, how should we write those rules?"</strong><br><br><strong>"In this example, we can observe that each term is 3 more than the previous term. In other words, we add three to the previous term to get the next term. Since the terms increase by a constant amount, the sequence is an </strong><i><strong>arithmetic sequence.</strong></i><strong>"</strong><br><br><strong>"To write the sequence as a recursive rule, we would write:</strong><br><br><strong>\(a_n\) = \(a_{n-1}\) + 3, </strong><i><strong>𝒏</strong></i><strong> &gt; 1</strong><br><br><strong>If we were to express the sequence as a close-form or explicit rule, we would write:</strong><br><br><strong>\(a_n\) = -9 + 3(</strong><i><strong>n</strong></i><strong> - 1).</strong><br><br><strong>Do you have any idea how this last formula was developed? Is there a general rule or structure for arithmetic sequences? If so, what is it?" </strong>Students should work in groups to find this generic shape. Distribute the Arithmetic Sequences resource (M-A1-2-3_Arithmetic Sequences).<br><br><strong>"Let's add another dimension to our analysis of this arithmetic sequence. Consider the following questions:</strong><br><br><strong>Does the sequence converge or diverge? What does it approach if it converges?</strong><br><strong>Does the total of the sequence converge or diverge? If the sum converges, what does it approach?</strong><br><strong>Do all arithmetic sequences follow the same pattern of behavior? Please provide an example."</strong><br><br><strong>Activity 7</strong><br><br>Use an Excel spreadsheet to help students identify an arithmetic sequence and illustrate its behavior as n increases.&nbsp;Their sequence could be linked to a real-world phenomenon or not.<br><br><strong>Geometric Sequences</strong><br><br><strong>Assume the sequence below is represented on the number line:</strong></p><figure class="image"><img style="aspect-ratio:505/87;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_59.png" width="505" height="87"></figure><p><strong>Here's what we should write on the board:</strong>&nbsp;1, \(1 \over 2\), \(1 \over 4\), \(1 \over 8\), \(1 \over 16\), \(1 \over 32\), …”<br><br><strong>“Now let’s write it using a table:</strong></p><figure class="image"><img style="aspect-ratio:238/511;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_60.png" width="238" height="511"></figure><p><strong>What is happening here? How can we characterize what occurs from term to term? Can we predict the future term based on the preceding term? Can we find the </strong><i><strong>n</strong></i><strong>th phrase without knowing the preceding one? If so, how should we write those rules?"</strong><br><br><strong>"In this example, each term is equal to \(1 \over 2\) of the previous one. To summarize, we increase or divide the previous term by \(1 \over 2\). Because the terms rise by a constant multiplier, </strong><i><strong>r</strong></i><strong> = \(1 \over 2\), the sequence is a&nbsp;</strong><i><strong>geometric sequence.</strong></i><strong>"</strong><br><br><strong>"If the sequence were to be expressed as a recursive rule, the following would be written: "</strong><br><br><strong>\(a_n\) = \(1 \over 2\)\(a_{n-1}\), </strong><i><strong>𝒏</strong></i><strong> &gt; 1</strong><br><br><strong>If we were to express the sequence as a close-form or explicit rule, we would write:</strong><br><br><strong>\(a_n\) = \(({1 \over 2})^{n-1}\)</strong><br><br><strong>Do you know how the last formula was created? Is there a general rule or structure for arithmetic sequences? If so, what is it?" </strong>(Students should attempt to identify this generic form.)<br><br><strong>"Let's add another aspect to our examination of geometric sequence. Consider the following questions:</strong><br><strong>Does the sequence converge or diverge? What happens if it converges?</strong><br><strong>Does the total of the sequence converge or diverge? If the sum converges, what is its approximate value?</strong><br><strong>Do all geometric sequences exhibit the same behavior? Please provide an example."</strong><br><br><strong>Activity 8</strong><br><br>Students should identify at least one important sequence. They must offer the sequence in a variety of forms and descriptions. For example, you may discuss convergence/divergence and how it applies to the sequence's context.<br><br><strong>Geometric Connections</strong><br><br>Give sequences the geometric relationship listed below:</p><figure class="image"><img style="aspect-ratio:530/89;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_61.png" width="530" height="89"></figure><p>Assign the students to finish the table below, write the diagonal sequences for the given polygons, and come up with a hypothesis for a rule that would explain the sequence.</p><figure class="image"><img style="aspect-ratio:595/204;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_62.png" width="595" height="204"></figure><p>Description of Pattern: _________________<br><br>Rule: ____________________________________<br><br><strong>"What if you compared the position number </strong><i><strong>n</strong></i><strong> to the number of diagonals? Would this change the rule? How?"</strong><br><br>Conduct a class discussion to review the lesson. Have each student respond to the following question: How does the study of patterns and sequences connect to the study of notation? Students should give an example.<br><br><strong>Extension:</strong><br><br>Once students have grasped the notion of a geometric portrayal of a sequence, ask them to develop a geometric representation of a sequence and identify at least one sequence from it. (For example, students may design a pool activity/representation or depict increasing triangles.)</p>