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Module 3 – Arithmetic and Geometric Sequences
3.1 Classroom Task: Growing Dots- A Develop Understanding Task Representing arithmetic sequences with equations, tables, graphs, and story context Ready, Set, Go Homework: Sequences 3.1 3.2 Classroom Task: Growing, Growing Dots – A Develop Understanding Task Representing geometric sequences with equations, tables, graphs, and story context Ready, Set, Go Homework: Sequences 3.2 3.3 Classroom Task: Scott’s Workout – A Solidify Understanding Task Arithmetic sequences: Constant difference between consecutive terms Ready, Set, Go Homework: Sequences 3.3 3.4 Classroom Task: Don’t Break the Chain – A Solidify Understanding Task Geometric Sequences: Constant ratio between consecutive terms Ready, Set, Go Homework: Sequences 3.4 3.5 Classroom Task: Something to Chew On – A Solidify Understanding Task Arithmetic Sequences: Increasing and decreasing at a constant rate Ready, Set, Go Homework: Sequences 3.5 3.6 Classroom Task: Chew On This – A Solidify Understanding Task Comparing rates of growth in arithmetic and geometric sequences Ready, Set, Go Homework: Sequences 3.6 3.7 Classroom Task: What Comes Next? What Comes Later? – A Solidify Understanding Task Recursive and explicit equations for arithmetic and geometric sequences Ready, Set, Go Homework: Sequences 3.7 3.8 Classroom Task: What Does It Mean? – A Solidify Understanding Task Using rate of change to find missing terms in an arithmetic sequence Ready, Set, Go Homework: Sequences 3.8 3.9 Classroom Task: Geometric Meanies – A Solidify and Practice Understanding Task Using a constant ratio to find missing terms in a geometric sequence Ready, Set, Go Homework: Sequences 3.9 3.10 Classroom Task: I Know . . . What Do You Know? – A Practice Understanding Task Developing fluency with geometric and arithmetic sequences Ready, Set, Go Homework: Sequences 3.10
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3.1Growing Dots A Develop Understanding Task
1. Describe the pattern that you see in the sequence of figures above.
2. Assuming the sequence continues in the same way, how many dots are there at 3 minutes?
3. How many dots are there at 100 minutes?
4. How many dots are there at t minutes?
Solve the problems by your preferred method. Your solution should indicate how many dots will be in the pattern at 3 minutes, 100 minutes, and t minutes. Be sure to show how your solution relates to the picture and how you arrived at your solution.
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3.1 Growing Dots – Teacher Notes A Develop Understanding Task
Purpose: The purpose of this task is to develop representations for arithmetic sequences that students can draw upon throughout the module. The visual representation in the task should evoke lists of numbers, tables, graphs, and equations. Various student methods for counting and considering the growth of the dots will be represented by equivalent expressions that can be directly connected to the visual representation. Core Standards: F-BF: Build a function that models a relationship between to quantities. 1: Write a function that describes a relationship between two quantities.*
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-LE: Linear, Quadratic, and Exponential Models* (Secondary I focus in linear and exponential only) Construct and compare linear, quadratic and exponential models and solve problems.
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Interpret expression for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context.
This task also follows the structure suggested in the Modeling standard:
Launch (Whole Class): Start the discussion with the pattern on growing dots drawn on the board or projected for the entire class. Ask students to describe the pattern that they see in the dots (Question #1). Students may describe four dots being added each time in various ways, depending on how they see the growth occurring. This will be explored later in the discussion as students write equations, so there should not be any emphasis placed upon a particular way of seeing the growth. Ask students individually to consider and draw the figure that they would see at 3 minutes
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(Question #2). Then, ask one student to draw it on the board to give other students a chance to check that they are seeing the pattern. Explore (Small Group or Pairs): Ask students to complete the task. Monitor students as they work, observing their strategies for counting the dots and thinking about the growth of the figures. Some students may think about the figures recursively, describing the growth by saying that the next figure is obtained by placing four dots onto the previous figure as shown:
Some may think of the figure as four arms of length t. with a dot in the middle.
Others may use a “squares” strategy, noticing that a new square is added each minute, as shown: As students work to find the number of dots at 100 minutes, they may look for patterns in the numbers, writing simply 1, 5, 9, . . . If students are unable to see a pattern, you may encourage them to make a table or graph to connect the number of dots with the time:
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Watch for students that have used a graph to show the number of dots at a given time and to help write an equation. Encourage students to connect their counting strategy to the equation that they write. For the discussion, select a student for each of the three counting strategies shown, a table, a graph, a recursive equation, and at least one form of an explicit equation. Discuss (Whole Group): Begin the discussion by asking students how many dots that there will be at 100 minutes. There may be some disagreement, typically between 100 and 101. Ask a student that said 101 to explain how they got their answer. If there is general agreement, move on to the discussion of the number of dots at time t. Start by asking a group to chart and explain their table. Ask students what patterns they see in the table. When they describe that the number of dots is growing by 4 each time, add a difference column to the table, as shown.
Time (Minutes) Number of Dots 0 1 1 5 2 9 3 13 … … t
Ask students where they see the difference of 4 occurring in the figures. Note that the difference between terms is constant each time. Continue the discussion by asking a group to show their graph. Be sure that it is properly labeled, as shown. Ask students how they see the constant difference of 4 on the graph. They should recognize that the y-value increases by 4 each time, making a line with a slope of 4.
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Now, move the discussion to consider the number of dots at time t, as represented by an equation. Start with a group that considered the growth as a recursive pattern, recognizing that the next term is 4 plus the previous term. They may represent the idea as: 𝑋 + 4, with X representing the previous term. This may cause some controversy with students that wrote a different formula. Ask the group to explain their work using the figures. It may be useful to rewrite their formula with words, like: The number of dots in the current figure = the number of dots in the previous figure + 4
This may be written in function notation as: 𝑓(𝑡) = 𝑓(𝑡 − 1) + 4. (You may choose to introduce this idea later in the discussion.) Next ask a group that has used the “four arms strategy” to write and explain their equation. Their equation should be: 𝑓(𝑡) = 4𝑡 + 1. Ask students to connect their equation to the figure. They should articulate that there is 1 dot in the middle and 4 arms, each with t dots. The 4 in the equation shows 4 groups of size t.
Next, ask a group that used the “squares” strategy to describe their equation. They may have written the same equation as the “four arms” group, but ask them to relate each of the numbers in the equation to the figures anyway. In this way of thinking about the figures, there are t groups of 4 dots, plus 1 dot in the middle. Although it is not typically written this way, this counting method would generate the equation 𝑓(𝑡) = 𝑡 ∙ 4 + 1. Now ask students to connect the equations with the table and graphs. Ask them to show what the 4 and the 1 represent in the graph. Ask how they see 4t +1 in the table. It may be useful to show this pattern to help see the pattern between the time and the number of dots:
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Time (Minutes) Number of Dots
0 1 1 1 5 1+4 2 9 1+4+4 3 13 1+4+4+4 … … t 1+4t
You may also point out that when the table is used to write a recursive equation like 𝑓(𝑡) = 𝑓(𝑡 − 1) + 4, you may simply look down the table from one output to the next. When writing an explicit formula like 𝑓(𝑡) = 4𝑡 + 1, it is necessary to look across the rows of the table to connect the input with the output. Finalize the discussion by explaining that this set of figures, equations, table, and graph represent an arithmetic sequence. An arithmetic sequence can be identified by the constant difference between consecutive terms. Tell students that they will be working with other sequences of numbers that may not fit this pattern, but tables, graphs and equations will be useful tools to represent and discuss the sequences. Aligned Ready, Set, Go Homework: Sequences 3.1
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Set Topic: Completing a table
Fill in the table. Then write a sentence explaining how you figured out the values to put in each cell. Explain how to figure out what will be in cell #8.
13. You run a business making birdhouses. You spend $600 to start your business, and it costs you $5.00 to make each birdhouse.
# of birdhouses 1 2 3 4 5 6 7
Total cost to build
Explanation:
14. You borrow $500 from a relative, and you agree to pay back the debt at a rate of $15 per month.
# of months 1 2 3 4 5 6 7
Amount of money owed
Explanation:
15. You earn $10 per week.
# of weeks 1 2 3 4 5 6 7
Amount of money earned
Explanation:
16. You are saving for a bike and can save $10 per week. You have $25 already saved.
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Go Topic: Good viewing window
When sketching a graph of a function, it is important that we see important points. For linear functions, we want a window that shows important information related to the story. Often, this means including both the x-‐ and y-‐ intercepts.
17. f(x) = -‐ 101 x + 1 18. 7 x – 3 y = 14
x: [ , ] by y: [ , ] x: [ , ] by y: [ , ]
x-‐scale: y-‐scale: x-‐scale: y-‐scale:
19. y = 3(x – 5) +12 20. f (x) = -‐15 (x + 10) – 45
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3.2 Growing, Growing Dots A Develop Understanding Task
At the At one minute At two minutes
beginning
At three minutes At four minutes
1. Describe and label the pattern of change you see in the above sequence of figures. 2. Assuming the sequence continues in the same way, how many dots are there at 5 minutes? 3. Write a recursive formula to describe how many dots there will be after t minutes. 4. Write an explicit formula to describe how many dots there will be after t minutes.
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3.2 Growing, Growing Dots –Teacher Notes A Develop Understanding Task
Purpose: The purpose of this task is to develop representations for geometric sequences that students can draw upon throughout the module. The visual representation in the task should evoke lists of numbers, tables, graphs, and equations. Various student methods for counting and considering the growth of the dots will be represented by equivalent expressions that can be directly connected to the visual representation. Core Standards: F-BF: Build a function that models a relationship between to quantities. 1: Write a function that describes a relationship between two quantities.*
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-LE: Linear, Quadratic, and Exponential Models* (Secondary Mathematics I focus in linear and exponential only) Construct and compare linear, quadratic and exponential models and solve problems.
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. c. Recognize situations in which one quantity grows or decays by a constant percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Interpret expression for functions in terms of the situation they model. 5. Interpret the parameters in a linear or exponential function in terms of a context.
This task also follows the structure suggested in the Modeling standard:
Launch (Whole Class): Start the discussion with the pattern of growing dots drawn on the board or projected for the entire class. Ask students to describe the pattern that they see in the dots (Question #1). Students may describe an increasing number of triangles being added each time or seeing three groups that each have an increasing number of dots each time, depending on how they
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see the growth occurring. This will be explored later in the discussion as students write equations, so there should not be any emphasis placed upon a particular way of seeing the growth. Ask students individually to consider and draw the figure that they would see at 5 minutes (Question #2). Then, ask one student to draw it on the board to give other students a chance to check that they are seeing the pattern correctly. Remind students of the work they did yesterday to write explicit and recursive formulas. These are new terms that should be reinforced at the beginning to clarify the instructions for questions 3 and 4. Explore (Small Group or Pairs): Ask students to complete the task. Monitor students as they work, observing their strategies for counting the dots and thinking about the growth of the figures. Some students may think about the figures recursively, describing the growth by saying that the next figure is obtained doubling the previous figure as shown:
𝑡 = 0 𝑡 = 1 Some may think of the figure as three groups that are each doubling.
𝑡 = 0 𝑡 = 1 t = 2 As students work to find the formulas, they may look for patterns in the numbers, writing simply 3, 6, 12, 24, 48 If students are unable to see a pattern, you may encourage them to make a table or graph to connect the number of dots with the time:
Time (Minutes) Number of Dots 0 3 1 6 2 12 3 24 4 48
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Watch for students that have used a graph to show the number of dots at a given time and to help write an equation. Encourage students to connect their counting strategy to the equation that they write. For the discussion, select a student for each of the counting strategies shown, a table, a graph, a recursive equation, and at least one form of an explicit equation. Have two large charts showing the dot figures prepared in advance for students to use in explaining their counting strategies. Discuss (Whole Group): Begin the discussion with the group that saw the pattern as doubling the previous figure each time. Ask them to explain how they thought about the pattern and how they annotated the figures. 3 6 12 Often, students who are using this strategy will think of the number of dots, without thinking of the relationship between the number of dots and the time. If they don’t mention the time at this point, be careful to point out the relationship with time when the next group presents a strategy that connects the time and the number of dots. Ask students to describe the pattern they see and record their words: Next figure = 2 × Previous figure Ask students to represent this idea algebraically. They may respond with expressions like: 2X where X is the number of dots in the previous figure. This is a chance to remind students that the instructions ask for a recursive formula that describes the number of dots at t minutes, which implies that the equation should be a function of t. If no students has written the equation in function form, help the class to understand how the equation 𝑓(0) = 3, 𝑓(𝑡) = 2𝑓(𝑡 − 1) expresses the idea that a way to find a term at time t is to double the previous term. Next, ask the group that saw this pattern of growth to explain the way they saw the pattern of growth.
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𝑡 = 0 𝑡 = 1 t = 2 Ask for a table to that shows the relationship between time and the number of dots. Ask students what patterns they see in the table. Ask students to add a difference column to the table, like they did in Growing Dots. Students may be surprised to see the difference between terms repeating the pattern in the number of dots. Ask students if they see a common difference between terms. Explain that since there is no common difference, it is not an arithmetic sequence.
Time (Minutes) Number of Dots 0 3 1 6 2 12 3 24 4 48
At this point, it can be pointed out that since you get the next term by doubling the previous term, there is a common ratio between terms. Demonstrate that: 6
3=
12
6=
24
12= 2
The common ratio between terms is the identifying feature of a geometric sequence, another special type of number sequence. Continue the discussion by asking a group to show their graph. Ask the class what they predict the graph to look like. Why would we not expect the graph to be a line? Be sure the graph it is properly labeled, as shown.
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Now, move the discussion to consider the number of dots at time t, as represented by an explicit equation. Ask a group to show their explicit formula for the number of dots at time t, which is: 𝑓(𝑡) = 3 ∙ 2𝑡 . Now ask students to connect the equations with the table and graphs. Ask them to show what the 2 and the 3 represent in the graph. Ask how they see 3 ∙ 2𝑡 in the table. It may be useful to show this pattern to help see the pattern between the time and the number of dots:
You may also remind students that when the table is used to write a recursive equation like 𝑓(0) = 3, 𝑓(𝑡) = 2𝑓(𝑡 − 1) you may simply look down the table from one output to the next. When writing an explicit formula like 𝑓(𝑡) = 3 ∙ 2𝑡, it is necessary to look across the rows of the table to connect the input with the output. Finalize the discussion by explaining that this set of figures, equations, table, and graph represent a geometric sequence. A geometric sequence can be identified by the constant ratio between consecutive terms. Tell students that they will continue to work with sequences of numbers using tables, graphs and equations to identify and represent geometric and arithmetic sequences. Aligned Ready, Set, Go Homework: Sequences 3.2
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Ready, Set, Go!
Ready Topic: Finding values for a pattern
1. Bob Cooper was born in 1900. By 1930 he had 3 sons, all with the Cooper last name. By 1960 each of Bob’s 3 boys had exactly 3 sons of their own. By the end of each 30 year time period, the pattern of each Cooper boy having exactly 3 sons of their own continued. How many Cooper sons were born in the 30 year period between 1960 and 1990?
2. Create a diagram that would show this pattern.
3. Predict how many Cooper sons will be born between 1990 and 2020, if the pattern continues.
4. Try to write an equation that would help you predict the number of Cooper sons that would be born between 2020 and 2050. If you can’t find the equation, explain it in words.
Set Topic: Evaluating Equations
Evaluate the following equations when x = { 1, 2, 3, 4, 5 }. Organize your inputs and outputs into a table of values for each equation. Let x be the input and y be the output.
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3.3 Scott’s Workout A Solidify Understanding Task
Scott has decided to add push-ups to his daily exercise routine. He
is keeping track of the number of push-ups he completes each day
in the bar graph below, with day one showing he completed three
push-ups. After four days, Scott is certain he can continue this pattern of increasing the number of
push-ups he completes each day.
1 2 3 4
1. How many push-ups will Scott do on day 10? 2. How many push-ups will Scott do on day n? 3. Model the number of push-ups Scott will complete on any given day. Include both explicit and
recursive equations. 4. Aly is also including push-ups in her workout and says she does more push-ups than Scott
because she does fifteen push-ups every day. Is she correct? Explain.
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Ready, Set, Go!
Ready Topic: Slopes between two points
Find the slope of the line that goes through each set of points.
1. (3,7) and (5, 10)
2. (-‐1, 4) and (3,3)
3. (0,0) and (-‐2, 5)
4. (-‐1, -‐5) and (-‐4, -‐5)
Set Topic: Finding terms for a given sequence
Find the next 3 terms in each sequence. Identify the constant difference. Write a recursive function and an explicit function for each sequence. (The first number is the 1st term, not the 0th). Circle the constant difference in both functions.
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Go Topic: Slope-‐Intercept Form
Write the equations in slope-‐intercept form.
7. 𝑦 = 12 + (𝑥 − 1)(−4)
8. !!6𝑦 + 9 = !
!(15𝑥 − 20)
9. !!21𝑦 + 7 = !
!(18𝑥 + 27)
Need Help? Check out these related videos: Finding slope http://www.khanacademy.org/math/algebra/ck12-‐algebra-‐1/v/slope-‐and-‐rate-‐of-‐change Writing the explicit equation http://www.khanacademy.org/math/algebra/solving-‐linear-‐equations/v/equations-‐of-‐sequence-‐patterns Writing equations in slope-‐intercept form http://www.khanacademy.org/math/algebra/linear-‐equations-‐and-‐inequalitie/v/converting-‐to-‐slope-‐intercept-‐form
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3.4 Don’t Break the Chain A Solidify Understanding Task
Maybe you’ve received an email like this before:
These chain emails rely on each person that receives the email to forward it on. Have you ever
wondered how many people might receive the email if the chain remains unbroken? To figure this
out, assume that it takes a day for the email to be opened, forwarded, and then received by the next
person. On day 1, Bill Weights starts by sending the email out to his 8 closest friends. They each
forward it to 10 people so that on day 2, it is received by 80 people. The chain continues unbroken.
1. How many people will receive the email on day 7?
2. How many people with receive the email on day n? Explain your answer with as many
representations as possible.
3. If Bill gives away a Super Bowl that costs $4.95 to every person that receives the email
during the first week, how much will he have spent?
Hi! My name is Bill Weights, founder of Super Scooper Ice Cream. I am offering you a gift certificate for our signature “Super Bowl” (a $4.95 value) if you forward this letter to 10 people. When you have finished sending this letter to 10 people, a screen will come up. It will be your Super Bowl gift certificate. Print that screen out and bring it to your local Super Scooper Ice Cream store. The server will bring you the most wonderful ice cream creation in the world—a Super Bowl with three yummy ice cream flavors and three toppings! This is a sales promotion to get our name out to young people around the country. We believe this project can be a success, but only with your help. Thank you for your support. Sincerely, Bill Weights Founder of Super Scooper Ice Cream
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Set Topic: Recursive and explicit functions of arithmetic sequences
Below you are given various types of information. Write the recursive and explicit functions for each arithmetic sequence. Finally, graph each sequence, making sure you clearly label your axes.
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Go Topic: Recursive and explicit functions of geometric sequences
Below you are given various types of information. Write the recursive and explicit functions for each geometric sequence. Finally, graph each sequence, making sure you clearly label your axes.
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Ready, Set, Go!
Ready Topic: Arithmetic and geometric sequences
Find the missing values for each arithmetic or geometric sequence. Then say if the sequence has a constant difference or a constant ratio, and say what the constant difference/rate is.
1. 5, 10, 15, ___, 25, 30… Constant difference or a constant ratio? The constant difference/ratio is __________.
2. 20, 10, ___, 2.5, ___... Constant difference or a constant ratio? The constant difference/ratio is __________.
3. 2, 5, 8, ___, 14, ___... Constant difference or a constant ratio? The constant difference/ratio is __________.
4. 30, 24, ___, 12, 6… Constant difference or a constant ratio? The constant difference/ratio is __________.
Set Topic: Recursive and explicit equations
Determine whether each situation represents an arithmetic or geometric sequence and then find the recursive and explicit equation for each.
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7. Time (days)
Number of Dots
1 3 2 7 3 11 4 15
Arithmetic or Geometric? Recursive: ___________________________________ Explicit: _____________________________________
8. Time (days)
Number of cells
1 5 2 8 3 12.8 4 20.48
Arithmetic or Geometric? Recursive: ___________________________________ Explicit: _____________________________________
9. Michelle likes chocolate but it causes acne. She chooses to limit herself to three pieces of chocolate every five days. Arithmetic or Geometric? Recursive: ___________________________________ Explicit: _____________________________________
10. Scott decides to add running to his exercise routine and runs a total of one mile his first week. He plans to double the number of miles he runs each week. Arithmetic or Geometric? Recursive: ___________________________________ Explicit: _____________________________________
11. Vanessa has $60 to spend on rides at the State Fair. Each ride cost $4. Arithmetic or Geometric? Recursive: ___________________________________ Explicit: _____________________________________
12. Cami invested $6,000 dollars into an account that earns 10% interest each year. Arithmetic or Geometric? Recursive: ___________________________________ Explicit: _____________________________________
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3.7 What Comes Next? What Comes Later? A Practice Understanding Task
For each of the following tables,
describe how to find the next term in the sequence, write a recursive rule for the function, describe how the features identified in the recursive rule can be used to write an explicit
rule for the function, and write an explicit rule for the function. identify if the function is arithmetic, geometric or neither
Example:
To find the next term: add 3 to the previous term Recursive rule:𝑓(0) = 5, 𝑓(𝑛) = 𝑓(𝑛 − 1) + 3 To find the nth term: start with 5 and add 3 n times Explicit rule: 𝑓(𝑛) = 5 + 3𝑛 Arithmetic, geometric, or neither? Arithmetic
x y 1 5 2 10 3 20 4 40 5 ?
n ?
x y 0 3 1 4 2 7 3 12 4 19 5 ?
n ?
x y 0 5 1 8 2 11 3 14 4 ?
n ?
Function A
1. To find the next term: _______________________________________
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3.8 What Does It Mean? - Teacher Notes A Solidify Understanding Task
Purpose: The purpose of this task is to solidify student understanding of arithmetic sequences to
find missing terms in the sequence. Students will draw upon their previous work in using tables
and writing explicit formulas for arithmetic sequences. The task will also reinforce their fluency in
solving equations in one variable.
Core Standards:
A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. Clusters with Instructional Notes: Solve equations and inequalities in one variable. Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16 .
Launch (Whole Class): Refer students to the previous work with arithmetic sequences. A brief
discussion of the chart that was created as part of the “What Come Next, What Comes Later” task
may be useful in drawing students’ attention to the role of the first term and the common difference
in writing explicit formulas.
Explore Part 1 (Small Group): Give students the task and be sure that they understand the
instructions. Tell them that they are looking for a strategy for solving these problems that works all
the time, so they should be paying attention to their methods. The task builds in complexity from
the problems on side one to the problems on side 2. Because the problems on the first side are all
asking students to find an odd number of terms, most students will probably average the first and
last terms to find the middle term, and then average the next two, and so on until the table is
completed. Some students may choose to write an equation and use it to find the missing terms.
Others may use guess and check to find the common difference to get to each term. Watch for the
various strategies and note the students that you will select for the discussion of the first part. Let
students work on the task until they complete the first page, and then call them back for a short
discussion.
Discuss (Whole Class): Begin the discussion by asking a student that used guess and check to
explain his/her strategy. Ask the class what understanding of arithmetic sequences is necessary to
use this strategy? They should answer that it relies on knowing when the common difference is
known, it can be added to a term to get the next term. This is using the reasoning that we use to
write recursive formulas. Next, ask the student that used an averaging method to explain his/her
method. Ask the class why this method works. Now ask the class what the advantages are to each
of these methods. Will they both work every time? If students haven’t noticed that all of these
examples are working with an odd number of missing terms, ask them what would happen if you
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3.9 Geometric Meanies – Teacher Notes A Solidify and Practice Task
Purpose: The purpose of this task is to solidify student understanding of geometric sequences to
find missing terms in the sequence. Students will draw upon their previous work in using tables
and writing explicit formulas for geometric sequences.
Core Standards:
A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. Clusters with Instructional Notes: Solve equations and inequalities in one variable. Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents, such as 5x = 125 or 2x = 1/16 . Launch (Whole Class): Explain to students that that today’s puzzles involve finding missing terms
between two numbers in a geometric sequence. These numbers are called “geometric means”. Ask
them to recall the work that they have done previously with arithmetic means. Ask, “What
information do you think will be useful for finding geometric means? Some may remember that the
first term and the common difference were important for arithmetic means; similarly the first term
and the common ratio may be important for geometric means.
Ask, “How do you predict the methods for finding arithmetic and geometric means to be similar?”
Ideally, students would think to use the common ratio to multiply to get the next term in the same
way that they added the common difference to get the next term. The may also think about writing
the equation using the number of “jumps” that it takes to get from the first term to the next term
that they know.
Ask, “How do you think the methods will be different?” Ideally, students will recognize that they
have to multiply rather than simply add. They may also think that the equations have exponents in
them because the formulas that they have written for geometric sequences have exponents in them.
Explore (Small Group): The problems in this task get larger and require students to solve
equations of increasingly higher order. Some students will start with a guess and check strategy,
which may work for many of these. Even if it is working, you may choose to ask them to work on a
strategy that is more consistent and will work no matter what the numbers are. Students are likely
to try the same strategies as they used for arithmetic sequences. These strategies will be successful
if they think to multiply by the common ratio, rather than add the common difference. If they are
writing equations, they may have similar thinking to this:
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5. Recursive Equation: Graph Explicit Equation:
Table
Create a context Janet wants to know how many seats are in each row of the theater. Jamal lets her know that each row has 2 seats more than the row in front of it. The first row has 14 seats.
Go Topic: Writing explicit equations
Given the recursive equation for each arithmetic sequence, write the explicit equation.