• Activity 2 will provide an opportunity to assess how in-depth the level of student comprehension is.  
• Use student responses during Activity 4 to determine how well students understand linear relationships.
• The Lesson 1 Exit Ticket (M-8-1-1_Lesson 1 Exit Ticket and KEY.doc) may be used to evaluate level of concept mastery for individual students.

Part 1: Nonnumeric Patterns

Explain to students, “A pattern involves the arrangement of a predetermined format of numbers, symbols, or objects. Patterns have been used to decipher ancient, unknown languages; predict the locations of planets in the galaxy at specific times; and program computers to become world-class chess players and even to learn from other human players. Basically, patterns come in all shapes and sizes and are incredibly powerful tools to analyze and predict.”

Display the pattern below to the class. Use counters, draw the pattern, or use a projection device.

“Is there a pattern to the squares?” (Yes.)

“What comes next?” (2 rows of 7 squares)

“Describe the pattern in words.” (A column of two squares is added to the right of each figure.)

Add the step numbers under each step of the pattern. Then ask students again to describe the pattern in words. Ask students to count the total number of squares in each step of the pattern. “How does the total number of squares relate to the step number?” (The total number of squares is always double the step number. For instance, there are 10 squares in Step 5.) “Let’s call the total number of squares in each step y. Let’s call the step number x. Using these variables, can you come up with an equation that relates the total number of squares and step numbers for all the figures in our pattern?” Through similar questioning, arrive at the rule for the pattern, y = 2x, or total number = 2 × step number. For additional practice with nonnumeric patterns, use Nonnumeric Patterns (M-8-1-1_Nonnumeric Patterns and KEY.doc) or use http://www.shodor.org/interactivate/activities/PatternGenerator/, an interactive Web site that creates nonnumeric patterns. After students finish the pattern, have them describe the pattern in words.

Activity 1

Challenge students to come up with one growing shape pattern. Then ask them to write the rule as an equation using the variables x and y.

Part 2: Numeric Patterns

“Let’s explore a few number patterns via sequences. A sequence is a list of numbers that may or may not follow a particular pattern. These sequences do have a particular pattern.” Write the following on the board:

• Sequence 1: −2, −5, −8, −11, …
• Sequence 2: 12, 15, 19, 24, …
• Sequence 3: 7, 15, 23, 31, 39, …
• Sequence 4: 1, 4, 9, 16, 25, …
• Sequence 5: 1, 8, 27, 64, …
• Sequence 6: 3, 9, 27, 81, …

“Which of these patterns have something in common? What do they have in common?”

Lead students to discover that the first and third sequences have something in common: they both have a constant difference between each term. We can say that this constant difference is a constant rate. The last sequence has a constant multiplier or ratio between each term. The other three sequences also have something in common: they do not have a constant difference (or constant rate) or constant multiplier between each term. In the second sequence, the change in difference between each subsequent term increases by 1 more than the previous amount added. The fourth sequence involves squares of the natural numbers. The fifth sequence involves cubes of the natural numbers. Remember, the natural numbers are the counting numbers, or 1, 2, 3,….

“If we plotted these values on a coordinate plane, with the x-values being 1, 2, 3, 4, …, respectively, what do you think the graphs would look like? To help us think about what the graph might look like let’s create an x–y chart using these x-values (1, 2, 3, 4) and the values from Sequence 1 from the list on the board (−2, −5, −8, −11). The values from the pattern will be the corresponding y-values.” Create an x/y chart similar to the one below on the board. While creating the x/y chart, explain to students the headings for each column. Then use a think-aloud to fill in the x/y chart.

“When we plot points on a coordinate plane, we need to have an x-value and a corresponding y-value. For the x-values, we are going to use 1, 2, 3, 4, so I will put those in the first column of the chart. For the y-values we will use the numbers in Sequence 1 which we looked at earlier: −2, −5, −8, −11. I will put those values in the second column of the chart. Then I can write the ordered pairs that will be used to plot on a coordinate plane.”

“By looking at the x/y chart, what do you predict the graph will look like for Sequence 1?” Students should predict a line (M-8-1-1_Displays 1-6.docx). Remind students that there is a constant rate of change as seen by the constant difference between the
y-values. Students can be asked to create x/y charts for the other patterns listed on the board, keeping the x-values at 1, 2, 3, 4, … for each.

For the remaining graphs, students should predict something close to a line for Sequence 2, a line for Sequence 3, a curve for Sequence 4, a steeper curve for Sequence 5, and a curve for Sequence 6. Lead students to state that two of the six are linear, whereas the other four are not linear, with linear simply representing a constant rate of change, or a line on a graph.

“The reason for imagining the look of the graph and the reason for plotting the graph is to look for the trend of the numbers and use that to make predictions or to develop a rule for the pattern. Remember that linear means a constant rate of change or a line on a graph. Knowing that a relationship is linear can be useful for identifying trends and making predictions in real-world situations.”

[Note: The point of having students think about the shape of the graph now is to allow them to imagine patterns as a graph without or prior to working with the equation form of a line. Doing so will help students truly make some of the big connections between patterns, functions, and graphs.]

• Sequence 1 (1, −2), (2, −5), (3, −8), (4, −11):

• Sequence 2: (1, 12), (2, 15), (3, 19), (4, 24)

• Sequence 3: (1, 7), (2, 15), (3, 23), (4, 31), (5, 39)

• Sequence 4: (1, 1), (2, 4), (3, 9), (4, 16), (5, 25)

• Sequence 5: (1, 1), (2, 8), (3, 27), (4, 64)

• Sequence 6: (1, 3), (2, 9), (3, 27), (4, 81)

“We can see that our predictions were correct simply based on the sequence of numbers we examined.”

“Now let’s explore some patterns in the forms of tables.”

Table A

“What do you notice about this pattern? As x increases by 1, what does y increase by?” (The y-values increase by 3.)

Table B

“What do you notice about this pattern? As x increases by 1, what does y increase by?” (For each change in x-value, the y-values are multiplied by 2.)

“What can we conclude from the two tables above?”

Allow students to think-pair-share, and then share their ideas. Once students share their ideas, emphasize the strategy of finding the differences in the y-values by doing a think aloud and modeling on the board referring to the table where necessary. “When I look at the y-values in Table A, I notice that the values increase by 3 each time. That is a constant amount. The
y-value 7 increases by 3 to 10, 10 increases by 3 to 13, and 13 increases by 3 to 16. Each subsequent term in the sequence is three more than the previous term. The difference in the consecutive y-values is constant; therefore, the relationship is linear. This constant difference also is the slope of the equation of the line. Slope is how steep a line goes up or down.

“Now look at Table B and see if you can identify a pattern like we just talked about.” Guide student understanding using questions similar to those listed below.

• “What are the values for y in Table B?” (2, 4, 8, 16, 32)
• “Is there a pattern in the y-values? If so what is it?” (Yes, you multiply the previous term by 2. The y-values increase by a constant multiplier, 2.)
• “Each subsequent term in the sequence is the ________ of the previous term and ____.” (product; two)
•  “Is the value of the difference between the y-values constant?” (No.)
• “Would the relationship be linear? Explain your reasoning.” (No, because the difference between each term is not constant.)

“Remember, if the difference in the consecutive y-values is constant, then the relationship is linear. The constant difference is also the slope of the equation of the line.

“Let’s look at Table A and Table B further. Are the constant increases between the
y-values in Table A and Table B to be considered the same? Did they increase in the same way? In other words, can we state another similarity? Do we notice an obvious difference?”

(The similarity between the two tables is that there is a constant change between the y-values for each change in x-value. The difference is that one change involves a constant amount added, whereas the other change involves a constant multiplier.)

“In Table A, we have what is called an arithmetic sequence. An arithmetic sequence is a sequence of numbers that increases by a constant amount, or involves a constant difference between terms. In Table B, we have a geometric sequence. A geometric sequence is a sequence of numbers that increases by a constant multiplier, or involves a constant ratio between terms.”

Give each student a copy of the Sequences worksheet (M-8-1-1_Sequences and KEY.doc). Have students determine how each sequence was formed. In other words, have them determine the constant difference between the subsequent numbers in each sequence and complete the second column of the chart. Model using the example given on the sheet if necessary. Have students discuss their findings with a partner. Monitor student performance and provide necessary support. Then as a class discuss the answers having students adjust answers where necessary. Post the Sequence Chart (M-8-1-1_Sequence Chart.doc) on the board for students to see or hand out a copy to each student to help clarify the terminology. Discuss the difference between an arithmetic sequence and a geometric sequence. Then have students identify with their partners whether each sequence on their Sequences worksheet is arithmetic or geometric and fill in the last column.

To assess understanding, give each student an index card. Have students write the word arithmetic on one side in large letters and geometric on the other side. Then show one of the sequences from the sheet and ask students to hold up the side of their index card that correctly identifies the type of sequence it is. Call on one student to explain his/her reasoning. Clarify misunderstandings with those students who may identify the sequence as the wrong type. If time permits or additional practice is needed, repeat the index card activity with sequences that students generate. Give partners a sheet of white paper and have them generate either an arithmetic or geometric sequence in 2–3 minutes and repeat the index card activity.

[Note: There are specific formulas to determine the actual rule used to find the value of the x^th term in a sequence, but that information goes beyond the scope of this lesson.]

“If we were to revisit the first table, we could work with the numbers and determine the rule that is used to find the value of the x^th term. We can do so at this point without using the arithmetic sequence algorithm (or the geometric sequence algorithm, as with the second table).”

Our table looked like this:

x	y
0	4
1	7
2	10
3	13
4	16
5	19
x

Activity 2

Consider a few possibilities for writing the rule for the table. Decide on a rule that fits the data in the table, showing how to find the x^th term without knowing the value of the previous term. Remind students to think about the strategies used earlier to find the rule for a pattern. Allow students time to explore with a partner. Monitor interaction and dialogue. Provide appropriate questioning to help jumpstart or guide thinking. Use questions similar to those listed below. If necessary, complete this as a whole group discussion using guiding questions.

• “Is this a sequence?
• “Is it linear? Explain your reasoning.
• “Do you see an obvious pattern or rule?
• “Does the rule have to work for each case listed in the table?” (Yes.)
• “What might the graph look like? Explain your reasoning.
• “If you look at the y-values are they getting larger or smaller?” (larger)
• “What operations can be used to get a larger number?” (add or multiply)
• “What operations can be used to get a smaller number?” (subtract or divide)
• “What difference do you notice in the first row?” (possibly add 6 or multiply by 7)
• “Does this work for the next row?” (No.)
• “What does this tell us?” (Answers will vary.)
• “Is more than one operation going to be necessary? Explain your reasoning.
• “Since the y-values are getting larger, what if I told you that you will need to use two operations and one of them is multiplication? Does that help you?”

Students should arrive at the following rule: . “Let’s replace our y-value for the x^th term with this rule, or formula.” Record this equation as the y-value on the table for the x^th term.

“Let’s work to find a rule that can describe how to find the value of the x^th term of the geometric sequence.”

“Can you find the rule for the sequence in this table? Think about the strategies you used with the previous example.” Allow students time to explore with a partner. Monitor student interaction and dialogue. Provide appropriate questioning to help jumpstart or guide student thinking without giving students the answer. Use questions similar to those listed below. Questioning can be done orally or using cards cut from the Jumpstart Questioning Cards (M-8-1-1_Jumpstart Questioning Cards.doc). Choose the appropriate card that may jumpstart a student’s thinking in a different way and place on the student’s desk. Walk away and monitor other students, and then revisit to see if students were able to progress further in their thinking with the Jumpstart Questioning Card.

• “Is this a sequence?
• “Is it linear? Explain your reasoning.
• “Do you see an obvious pattern or rule?
• “Does the rule have to work for each case listed in the table?
• “What might the graph look like? Explain your reasoning.
• “If you look at the y-values are they getting larger or smaller?
• “What operations can be used to get a larger number?
• “What operations can be used to get a smaller number?
• “What difference do you notice between the x-values and y-values?
• “What does this tell us?
• “Is more than one operation going to be necessary? Explain your reasoning.
• “Since the y-values are getting larger, what if I told you that you will need to use exponents? Does that help you?
• “How does the exponent link to the term number?”

Students should arrive at the following rule: y = 2^x. “Let’s replace our y-value for the x^th term with this rule, or formula.” Record this equation as the y-value on the table for the x^th term.

“What is the eighth term in the pattern?” Have students work on figuring out the eighth term, and then ask the class, “Did any of you find a way to figure out the eighth term without figuring out the sixth term first?” If students did, ask them to explain their thinking to the class. If no student did, ask students, “How do you find the second term in the pattern?” (Multiply the first term by 2.)

In response to this, say, “I’m going to rewrite the pattern over here, but write it in a slightly different form. First, I’m going to start with 2, but then, for the second term, instead of writing 4, I’m going to write 2 × 2, since to get to the second term, I just multiply the first term by 2. How do I get from the second term to the third term?” (Multiply the second term by 2.) Rewrite the third term as 2 × 2 × 2. “Let me create another column in my table to help with this illustration.” Repeat this for the fourth term, and then tell students, “I’m getting tired of writing all these twos and multiplication signs. Is there an easier way for me to write something like 2 × 2 × 2 × 2?” (2⁴) Rewrite the entire pattern using a base of 2 and increasing exponents. “Let me create another column, writing the expression with an exponent.”

Now, ask students, “So what is the third term in our pattern using powers?” (2³) Emphasize the word third in both the question and the answer. Repeat the question for the fourth term, again emphasizing the word fourth. Ask what the eighth term is using exponents. (2⁷) Ask about a couple of other terms, including some very large terms, such as the 25^th or 375^th term.

“Finding a general, quick way to represent terms is often very useful in patterns. If we have to find the 375^th term, knowing that it is 2³⁷⁵ is a lot quicker than having to find all
374 terms that come before it.

“What if we wanted to find the x^th term, but I don’t tell you what the x stands for? How do you find the x^th term in this pattern?” If students have difficulty, repeat some of the previous questions, verbally emphasizing both the term number and the exponent. Repeat a few of these, and ask what the x^th term will be. Students should make the leap and recognize that the term number and the power are the same, and so the x^th term will be 2^x. “Let me write that formula in our table, .

“What do we notice about the two patterns? Are there differences? Are there similarities?” (The first formula includes a power of 1 on the term involving the variable. The second formula sometimes includes exponents greater than 1.) “What does the exponent tell us?

“Notice the first formula, , has a 3 in front of the n. This number is the coefficient. The pattern had a constant increase of 3 and is represented in the formula as the coefficient. Also note that the 4 in our formula is the y-value when x = 0. These recognitions will be important as we explore linear functions later on. Notice the second formula, , has a constant multiplier of 2 included. This number is the constant multiplier in the second pattern.”

Activity 3

“Here is a question to think about: If both patterns change by a certain amount, are they both linear? In other words, if we graphed the points as ordered pairs on a coordinate plane, would the resulting graph be a line or some other shape?” Have students discuss ideas with a partner, citing examples. Students should realize that the first graph is a line, whereas the second graph is nonlinear. The constant change, whether it is a difference or a ratio, does matter when determining the graph of the pattern.

Activity 4

“Now that we’ve examined the geometric sequence in a structured form, I would like you to look at another arithmetic sequence in the same structured manner.

“Consider the following arithmetic sequence: 2, 5, 8, 11,…

“Work with a partner to create a table that helps with the process of determining the rule for the x^th term. You will determine the rule for the sequence and show your steps and reasoning.” Reconvene as a class and fill in the table below. A partially finished table is shown below.

“Now, let’s go in the opposite direction and create a pattern from a given rule. We can list the terms of the pattern from the rule.

“To find the first term in the pattern represented by 2x + 1, remember that x represents the term number. If we want to find the first term, then x = 1. To find out what the first term is, we just substitute 1 wherever we see x. What do you get for the first term when
x = 1?” (3)

“For the second term, we use x = 2. What is the second term in this pattern?” (5)

[Note: Record students’ comments in a table similar to the one shown below. This is simply a model. You will want to include more than just the values shown.]

x	y
1	3
2	5
300	601
x

Remind students that the advantage of having the general form of the x^th term, like 2x + 1 is that, if you want to find the 300^th term, you don’t have to find the 299 terms that come before it. You can just substitute in 300 for x and get the answer you want. “What is the 300^th term?” (601)

Activity 5

Have students work in pairs. Each student should come up with a pattern, with creativity encouraged. Students will trade papers and find the rule for the x^th term in their partner’s pattern. Have students trade papers back and check each other’s work.

Students should walk away with the following understanding thus far:

• Sequences are used to represent patterns.
• A pattern can be linear or nonlinear. A graph can be used to illustrate a pattern.
• A pattern with a constant difference is linear. The constant difference in a linear pattern is also the constant rate of change and coefficient in the formula representing the pattern.
• A pattern with a constant multiplier is not linear. The constant multiplier is shown in the formula representing the pattern as the constant.
• A rule can be written to describe a pattern and find the value of the x^th term. A pattern can also be generated from a general rule.

Activity 6

• Give students several sequences and ask them to determine the rule used to find the x^th term.
• Give students several rules and ask them to find the 5th, 10th, 100th, 500th, etc., terms in the sequence.
• Spend time looking through some magazines, books, newspapers, and any other reputable texts and find examples of numerical patterns. Study the patterns and choose one linear pattern to discuss. Provide a rule for the linear pattern. Discuss the rate of change, and represent the pattern in tabular and graphical form. Finally, discuss what the trend shows in the context of the data represented.

Have students complete Lesson 1 Exit Ticket (M-8-1-1_Lesson 1 Exit Ticket and KEY.doc).

Extension:

Use the suggestions below to tailor the lesson to meet the needs of the students.

• Routine: Periodically throughout the school year, have students practice with numeric linear patterns from this Web site:

http://harmon-middle-school.wikispaces.com/Linear+and+Nonlinear+numerical+patterns

• Small Groups: Students who would benefit from additional practice may be assigned to work with the nonnumeric linear patterns in the pdf document found here:

http://mrparina.com/yahoo_site_admin/assets/docs/15_Notes.26411240.pdf

• Expansion: Students who are ready for a challenge beyond the standard may be given this activity. Consider the x/y chart below:

Students should be given time to explore the possibilities. This pattern is a lot harder, so give students enough time to work to discover the formula, but not so long that they become frustrated. The formula, y = (−2)^x, can be used to find the value of the n^th term in the sequence or pattern. Students should replace the y-value for the x^th term with the formula found above.

Encourage students to plot the points in the second graph to make the distinction (see below). Students should notice the sporadic plotted points in the graph. A nice discussion related to the multiplication by a negative number and result on the graph can ensue.

Give students drawings of polygons (M-8-1-1_Polygon Expansion Activity and KEY.doc). Have students find the total number of diagonals possible in each polygon and look for a rule that can be used to determine the total number of diagonals. Remind students that a diagonal is a line segment joining two vertices of a polygon, not including the line segments that form the sides of a polygon. Have students create their own graphic organizer to represent their findings from this activity.

[Note: The number of diagonals from a single vertex of a polygon is three less than the total number of vertices (d = v − 3, where d = number of diagonals and v = number of vertices of a polygon). The total number of diagonals possible in a polygon can be determined with the following rule: , where v is the number of vertices of a polygon and d is the total number of diagonals.]