name: breanna andrist -...

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Name: Breanna AndristGrade Level: Algebra 1Lesson Title: Lesson 7.1 Recursive Routines

Content Big Idea and Learning Targets Content Big Idea:

o Modeling exponential growth through recursion using constant multipliers Learning Targets:

o By the end of this lesson, students should be able to complete the following: Calculate the constant multiplier in a sequence and use it to find

additional terms. Write an equation for exponential growth given the starting value and

constant multiplier Solve problems that can be represented by exponential equations. Identify the differences between linear and exponential equations.

Why are these important?o By the end of this lesson, students should understand the differences between

linear and exponential functions and be able to write equations for exponential models. Both the lesson and practice opportunities allow students to see how exponential models are used in the real-world. The investment practice problem is also a practical example that students will experience outside of the classroom.

How do the targets and big ideas relate to your discipline and to your specific students?o Students have already been exposed to linear equations (graphs, equations,

tables and examples) but this lesson activates their prior knowledge in order to understand exponential equations and the differences from linear models.

o Students are able to work with a partner on the investigation and practice worksheet and they will also have opportunities to demonstrate work and understanding in front of their peers. Because my students are social and like to talk through problems, this lesson allows them opportunities to engage with each other several times over the course of the period.

State Standards Addressed:o A1.7.C: Express arithmetic and geometric sequences in both explicit and

recursive forms, translate between the two forms, explain how rate of change is represented in each form, and use the forms to find specific terms in the sequence.

o A1.1.E: Solve problems that can be represented by exponential functions and equations.

Academic Language Key vocabulary:

o Geometric Sequence : A series of terms generated by multiplying or dividing by a constant is known as a geometric sequence.

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o Constant Multiplier : In a sequence that grows or decreases exponentially, the number each term is multiplied by to get the next term. The value of (1 + r) in the exponential equation y = A(1 + r)x.

o Exponential Growth : A growth pattern in which amounts increase by a constant percent. Exponential growth can be modeled by the equation y = A(1 + r)x where A is the starting value, r is the rate of growth written as a decimal or fraction, x is the number of time periods elapsed, and y is the final value.

Assessment of Student Learning How will you assess student learning?

o Formative Assessment: Students will be asked several questions throughout the lesson to informally assess their understanding of learned concepts (key vocabulary and differences between linear and exponential functions). In addition to class discussions, teacher will also walk around room to observe student work as a means of informal assessment. Formative assessment will be done using the lesson worksheet as well as the practice worksheet. Both of these will allow an opportunity to assess comprehension.

o Student Voice: Students will have opportunities to express their current understanding of recursive routines and exponential models prior to starting the lesson. At the end of the lesson (in the practice worksheet) students will have an opportunity to express their learning from the lesson, how exponential models are used in the real-world, and why it is important to understand what they represent.

o Accommodations/Modifications: Teacher will walk around the room during the lesson to help assist students who continue to struggle with concepts. Several students will require additional explanations because of the complexity of the questions. Students will also be allowed to work with a partner to help work through problems.

Instruction and Engagement to Support Student Learning (Lesson Breakdown) Supplies/Materials :

o Attached Worksheets (Warm-Up, Lesson 7.1 and Practice 7.1) Warm-Up Activity (see attached warm-up activity)

o Warm-up activity is designed to activate prior knowledge of patterns that are formed by continuously adding the same number to create a recursive routine (arithmetic sequences). This warm-up is designed to activate prior knowledge of linear equations and sequences in order to understand the differences between linear and exponential equations.

Define Lesson Target : Transitioning from linear equations to exponential equations. Students will be identifying common ratios in geometric sequences, which are used in exponential equations. Students will also be able to apply understanding of exponential equations to a real-world situation involving exponential growth.

Worksheet Lesson 7.1 Recursive Routines (see attached)

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o Ask students to write down what they currently know about exponential models or recursive routines in the space provided in the worksheet.

o Class discussion about why they think exponential models are important or seen in real-world situations.

o Activate prior knowledge by starting a linear vs. exponential poster on the board (includes equation, rules, table of values and graph of function)

o Introduce Bug investigation by using picture flip chart to compare linear growth and exponential growth.

o Bugs, Bugs, Everywhere Investigation (to be completed in pairs): Discuss the situation in the problem and explain the table Students complete steps #1 and #2 (teacher walk around room) Ask student to come up to board with table information and class

discussion about steps #1 and #2 Students complete steps #3 - 6 after brief explanation Ask student to come up to board with graph and class discussion about

steps #3 – 6. Add graph of exponential equation to linear vs. exponential poster.

Teacher discussion of general exponential equation, the bug exponential equation (including expanded form) and a general equation for growth and decay.

Students complete step #7 and student puts answer on board after work time. Continue class discussion about exponential models and further questions (including any additions to linear vs. exponential poster).

Worksheet: Practice Lesson 7.1 Recursive Routines (see attached) o Practice objective: worksheet is designed to have students practice finding the

constant multiplier of a geometric sequence and to apply the concepts to a real-world situation involving investments.

o Pass out worksheet to students and show #1 as example on the board.

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o Allow students time to complete some of worksheet in class and the remainder will be homework due the following day.

o The end of the worksheet has opportunity to evaluate the lesson and ask any further questions, remind students to complete both questions.

Closing the Lesson o Ask students what they learned today about exponential models and the

differences between linear and exponential equations. o Tell students that the next day they will be doing two hands on activities that

involve taking data and writing an exponential equation to fit a set of data. The activities will also demonstrate the differences between exponential growth and decay.

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Warm-Up Lesson 7.1 Recursive Routines

NAME: _________________________________________ PERIOD: __________

Find the missing values in each of the following sequences.

1) 5, 10, 15, ____, 25, 30, _____, 40a. What rule did you use for the above sequence? ____________________



NAME: _________________________________________ PERIOD: __________





NAME: _________________________________________ PERIOD: __________




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Lesson 7.1 Recursive Routines

NAME: _________________________________________ PERIOD: __________

Lesson Objectives: Begin to investigate geometric sequences using recursive routines. See examples of how exponential equations are used to model real-life growth

being simulated by recursive routines.

What I currently know about recursive routines and exponential models: ____________________________________________________________________________________________________________________________________________

Lesson Vocabulary and Equations:

Geometric Sequence:

Constant Multiplier:

Exponential Equation (general):

Exponential Growth and Decay:

Bugs, Bugs, Everywhere Investigation:

Imagine that a bug population has invaded your classroom. One day you notice 16 bugs. Every day new bugs hatch, increasing the population by 50% each week. So, in the first week the population increases by 8 bugs.

Step 1: In the table below, record the total number of bugs at the end of each week for 5 weeks.

Weeks Elapsed Total # of BugsIncrease in # of bugs

(rate of change per week)

Ratio of this week’s total to last week’s total

Start (0) 16

1

2

7

3

4

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Step #2: The increase in the number of bugs each week is the population’s rate of change per week. Calculate each rate of change and record it in the table. Does the rate of change show a linear pattern? Why or why not? _________________________________________________________________________________________________________________________________

Step #3: Calculate the ratio of the total number of bugs each week to the total number of bugs the previous week and record it in the table (example: divide the population after 1 week has elapsed by the population when 0 weeks have elapsed). How do these ratios compare? ______________________________________________________________________________________

Step #4: Let x be the number of weeks elapsed, and let y represent the total number of bugs. Graph the data using (0, 16) for the first point in the graph below:

Step #5: Does the graph appear to be linear, why or why not? __________________________________________________________________________________________________________

Step #6: What do you notice about the slope of the line segments when you go from week 0 to week 5? ____________________________________________________________________________________________________________________________________________________

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Step #7: Using the equation above, how many bugs would there be in 20 weeks? How many bugs would there be in 30 weeks? ________________________________________________

General Exponential Equation

Exponential Form Example

Expanded Form Example

Exponential Growth/Decay

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Practice: Lesson 7.1 Recursive Routines

NAME: ___________________________________________ PERIOD: _________

Practice Objective: Continue to investigate geometric sequences using recursive routines, by recognizing starting values and constant multipliers. Students will also be able to apply these concepts to a real-world example about investments, which uses the exponential growth equation.

For each sequence below: state the starting value, the constant multiplier, and the next terms in the sequence.

1) 16, 24, 36, 54, 81, _____, _______

Starting Value: ___________

Constant Multiplier: ____________

2) 4, 8, 16, 32, 64, _____, _______



3) 20, 32, 51.2, 81.92, 131.072, _____



4) 3, 3.9, 5.07, 6.591, ______



5) 27, 18, 12, 8, _____



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Real-World Application

Maria has saved $10,000 from her part-time job and wants to invest it for college. The plan she chose to invest her money in, grows by 5% each year. Complete the table below showing how the investment grows year by year.

Year Current BalanceInterest

(balance x interest rate)

New Balance

(current balance + interest)

1 $10, 000 $10,000 x 0.05 = $500 10,000(1 + 0.05) or $10,500

2 $10,500

3

4

5

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Starting Value for the Investment: ________________

Constant Multiplier: ___________

Factored Form of Constant Multiplier: _______________

Exponential Equation for Investment: ________________

Balance of investment in 20 years: ___________________

What I have learned from this lesson that I did not know before: ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

What questions do I still have about exponential equations or recursive routines: _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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