Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 1: Identify parent functions by their algebraic representations and graphs, and construct
geometric and algebraic representations of their transformations.
Standard:
SMP:
F.BF.3
5, 7
Reactivate students’ background knowledge about the “parent” functions they have studied so far by
distributing one card to each student containing a function name, graph, or algebraic representation
from the resource Matching Activity. Make and distribute some duplicate cards or reduce the number
of function types as needed. Provide time for students to form groups of three (or four, if there are
duplicates) by finding the matching cards from each category. As students search for matches, circulate
to ensure that students have grouped themselves accurately, and assist as needed. Ask:
 What are the key features of each function family that help you to distinguish their graphs (e.g.,
Trigonometric functions are periodic. Exponential functions have horizontal asymptotes,
logarithmic functions have vertical asymptotes, and rational functions can have both.)?
 What characteristics of the algebraic representations impact the graphs (e.g., When rational
functions have variables in the denominator, the graphs will have vertical asymptotes. The
parent functions of absolute value functions and radical functions with even roots will stay
above the x-axis.)?
With students seated in these same groups, reactivate students’ background knowledge of function
transformations and their geometric and algebraic representations by presenting the resource
Transformations (Transformations Sample Answers), and prompt students to complete the tables using
image 1 of #1 as a model. Ask:
 What has changed in each image point and how are these changes reflected in the graphs and
the algebraic representations (e.g., A change to the x-coordinates results in a horizontal
translation or dilation and a substitution for the independent variable x. A change to the ycoordinates results in a vertical translation, dilation, or reflection and a corresponding change to
the dependent variable y.)?
 Why do you add 3 to the independent variable in the algebraic representation when the graph is
translated 3 units to the left (e.g., When an image point is translated 3 units to the left from the
pre-image, 3 is subtracted from the x-coordinate, so 3 must be added to the new x-coordinate to
align to the same y-coordinate.)?
Identify students who can explain horizontal transformations and invite them to share their thinking,
then call on students randomly to share and explain their group’s answers to selected problems.
Facilitate a discussion of any differences in opinions between groups.
Ask: How does identifying the key features of a graph help you to construct its algebraic
representation?
Additional Resources for Two-year Algebra 2:
Use the resource Modeling with Functions (Modeling with Functions Sample Answers) to engage
students in additional explorations of transformations of a variety of parent functions. Encourage
students to use technology as appropriate, and ensure that students can explain how changes to the
algebraic representations impact the graphs.
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to construct the algebraic representation of a function given the key features
of its graph.
Note to Teacher:
 In SLT 3 of this unit, students will be re-introduced to the Modeling Cycle, which will provide
structure for all subsequent SLTs in the unit. Because real-world problems often require a
model that is a combination and/or transformation of familiar functions, this SLT sets the stage
with a review of parent functions and their transformations. In SLT 2, students will explore
sums and products of parent functions. The Modeling Cycle will be revisited in Pre-calculus as
students further extend their understanding of functions as models to include compositions of
parent functions.
 The teacher will need to prepare in advance the resource Matching Activity by printing onesided, cutting along the solid lines, and laminating or gluing to cards. Each student in the class
should have exactly one card, so a reduction in the number of functions used or duplication of
some cards may be necessary to match the number of students in the class.
Additional Resources:
Desmos: Marbleslides-periodics
Desmos: Marbleslides-exponentials
Desmos: Marbleslides-rationals
MAP: Representing Functions of Everyday Situations
cK–12 Flexbook Unit 4 Topic 1 SLT 1 pdf
cK–12 Flexbook Unit 4 Topic 1 SLT 1
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 2: Analyze the graphs of combinations of functions.
Standard:
SMP:
F.BF.1b
4, 7
Present pages 1–2 of the MVP resource Imagineering (Imagineering Teacher Notes, Imagineering
Sample Answers). Provide time for students to read the context, and explain that they will be trying to
imagine the graphs of the equations based on the key features of the parent functions they see
represented. After students have considered the given functions individually, provide time for them to
discuss their ideas with a partner, and then facilitate a whole-group discussion as outlined in the
teacher notes. Consider providing partners with poster paper or individual white boards to display their
graphs. Ask:
 Why do all of the graphs have to be functions (e.g., The equations all give exactly one height of
the rider above the ground for each distance from the start of the ride.)?
 What is the impact to the graph of adding another function to a sine function (e.g., Since the
output values are added together for a given input value, the added function should end up
being the new “midline”.)?
 What is the impact to the graph of multiplying a sine function by another function (e.g.,
Multiplying the sine function by another function would “sandwich” the new function between
the multiplied function and its negative, because the sine function always stays between –1 and
1.)?
 What is the impact to the graph of taking the absolute value of a sine function (e.g., Since an
absolute value function always has output values greater than or equal to 0, the new function
should only have values greater than or equal to 0. Since negative function values will become
positive, parts of the graph below the x-axis should be reflected across the x-axis.)?
Once students have had adequate time to debate their ideas about the graphs, present pages 3–4 of
Imagineering and provide time for them to use technology to graph the functions. If students need
additional support to locate the key features of the graphs, consider giving them the windows
suggested in the teacher notes or Imagineering Graph Paper, which has the suggested windows built in.
Provide time for students to compare the actual graphs to their conjectures and analyze the similarities
and differences, with the goal of improving their ability to “see” the graphs of combined function
equations. Encourage students to apply their learning by developing an equation for their own ride,
exchanging with a partner to conjecture a graph, and comparing the conjecture to the actual graph.
As students continue to develop their intuition for the graphs of combined functions, provide time for
students to complete the resource Matching - Formulas & Graphs (Matching - Formulas & Graphs
Sample Answers) with their partners. Consider printing the equations one-sided and providing students
with scissors to cut out the graphs if it is helpful for them to move the graphs between the equations.
Ask: How do the features of parent functions impact the graph when their equations are changed by
adding or multiplying the functions or taking the absolute value of a function?
Additional Resources for Two-year Algebra 2:
Use the resource Modeling with Functions (Modeling with Functions Sample Answers) to engage
students in a more scaffolded exploration of combined functions and their graphs.
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to describe the features of the graph of a combined function obtained by
adding, multiplying, or taking the absolute value.
Note to Teacher:
 In SLT 3 of this unit, students will be re-introduced to the Modeling Cycle, which will provide
structure for all subsequent SLTs in the unit. Because real-world problems often require a
model that is a combination and/or transformation of familiar functions, SLT 1 set the stage
with a review of parent functions and their transformations. In this SLT, students explore sums
and products of parent functions. The Modeling Cycle will be revisited in Pre-calculus as
students further extend their understanding of functions as models to include compositions of
parent functions.
 Print Matching - Formulas & Graphs one-sided, and consider cutting out the graphs or
providing students with scissors to cut them out, so that students can move the graphs as they
consider the given equations.
 Students are encouraged to use technology appropriately in this lesson. They should be able to
analyze the algebraic representation of a function to choose a meaningful window and identify
key features of the graph, all of which will inform the suitability of the graph they obtain using
technology.
Additional Resources:
Functions Game (Functions Game Sample Answers)
NRich: Picture the Process I
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 3: Understand how the modeling cycle can be used to solve real-world problems.
Standard:
SMP:
N.Q.2, A.CED.1
4, 5
Present pages 1–4 of the Hybrid Cars flipchart and provide the resource Vehicle Information 1. Allow
time for students to write what they notice and wonder about the information they have been presented
with, then prompt pairs of students to share their responses with each other. Select students to share
their responses with the whole class, capturing their responses on the board. Ensure that the discussion
includes the following key points:
 I notice that the hybrid models cost more than the standard models.
 I notice that the hybrid models get better gas mileage than the standard models.
 I wonder if the amount of money a car buyer would save on gas would make up for the higher
price of the hybrid car.
Ensure that students are able to make sense of the information about gas mileage presented in the ads
shown in Vehicle Information 1. Page 5 of the flipchart provides information from the 2016 Fuel
Economy Guide (published by the EPA) that explains how the fuel economy estimates are determined.
Ask students to share some of the ways that hybrid cars save gas. If time allows, present page 6 of the
flipchart and follow the link to the flash animation that provides information about how hybrid cars
work.
Present page 7 of the flipchart, which frames the question that drives all of the subsequent activities.
Provide time for students to think about this question and to formulate a plan for addressing it. Explain
that mathematical modeling provides a useful structure for addressing complex questions like this one.
Present page 8 and provide time for students to think about the nature and purpose of mathematical
modeling. Select students to share their responses with the class. Ensure that the discussion includes
the following key points:
 Modeling with mathematics is a process that helps you make good decisions. The process
includes:
o identifying essential quantities and working to understand their relationships,
o choosing appropriate tools to represent those relationships, and
o drawing conclusions and relating them to the decision at hand.
Present page 9 of the flipchart and ask students to discuss what they think happens at each stage of the
modeling cycle. Present the resource The Modeling Cycle and provide time for students to read about
the process of modeling with mathematics. Be sure to discuss the importance of choices, assumptions,
and approximations in the modeling process.
Present Hybrid Cars Capture Sheet to students, then present page 10 of the flipchart. Provide sufficient
time for students to respond to question #1 on the handout. Select students to share their responses with
the class, capturing their responses on the board. Of all the quantities that are involved in this situation,
which do you think are most essential (e.g., the price of each car and the annual cost of fuel)?
Present page 11 of the flipchart. Ask:
 What branches of mathematics should you consider (e.g., algebra, geometry, and statistics)?
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7


What are some ways to represent the relationship between two or more quantities (e.g., We can
use words, tables, graphs, or symbols.)?
What technological tools can help with your analysis (e.g., graphing utilities, spreadsheets,
computer algebra systems, and dynamic geometry software)?
Provide time for students to respond to #2 on the handout. Select students to share their responses with
the whole class, capturing their responses on the board. Ensure that students consider the relative
merits of modeling the problem using words, tables, equations, and graphs, and that they consider
ways that technology could be useful to them.
Present the Exploring Assumptions flipchart. Page 1 restates the question and is provided for
convenience. Prompt students to form small groups, then present each group with exactly one page
from the resource Vehicle Information 2. Provide time for students to respond to #3–4 on Hybrid Cars
Capture Sheet. For each pair of vehicles, select a student to display the group’s work and explain their
reasoning to the class. Click the image on page 2 of the flipchart and use the dropdown menu on the
web site to select a hybrid vehicle. Prompt students to compare their results with those produced by the
online tool. Be prepared to discuss any slight differences that arise and the reasons for those
differences. In particular, note that the MSRP data shown at fueleconomy.gov is slightly different from
the information shown in Vehicle Information 2.
Explain that by examining the assumptions that are present in the initial model, we may find places
where we can refine the model to make it more useful, at a cost of added complexity. Ask:
 In our analysis so far, what assumptions have we made (e.g., We assumed that the 2016 Fuel
Economy Guide provided an accurate estimate for the cost of fuel.)?
 What factors affect the annual cost of fuel (e.g., the number of miles driven in a year, the fuel
economy of the vehicle, and the price of gas)?
Present page 3 of the Exploring Assumptions flipchart. Ask: what assumptions did the authors of the
guide make (i.e., A typical driver travels 15,000 miles per year, she drives 55% of the time under city
driving conditions and 45% of the time under highway conditions, and fuel costs $1.86 per gallon for
regular unleaded gasoline.)? Prompt students to calculate the annual cost of fuel using the “Combined
MPG” figure supplied in Vehicle Information 2, then to compare their calculation to the annual fuel
cost as shown. Be prepared to discuss minor differences in the results and why they arise.
Present pages 4–7 of the flipchart. As each page is presented, ask students to consider how they might
adjust their model in light of the information shown. Engage the class in a whole-group discussion,
emphasizing the following key points:
 Page 4 shows that there is significant variation in the number of miles people drive in a given
year. For instance, a typical 65-year-old female travels about 4800 miles per year, so it would
take her longer to benefit from cost savings on gas as compared to a driver who travels 15,000
miles per year.
 Page 5 shows that the price of gas in Maryland has varied significantly in the last 5 years, rising
to as much as $4.00 per gallon. Drivers of hybrid vehicles benefit more when gas prices are
high than when gas prices are low.
 Page 6 shows that people typically own new vehicles for 6–7 years. If it takes longer than that
to benefit from savings on gas, it may not be worth it to purchase a hybrid model.
 Page 7 shows that modern vehicles can last 200,000 miles or more. This means that a car that
travels 15,000 miles per year has a lifespan of about 13–14 years. We can use this information
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
to determine how much money we could save if we planned to own the vehicle as long as
possible.
Present page 8 of the flipchart. Prompt students to use the space under #5 to respond to the question
shown on the flipchart, then return to the web site linked on page 2 and click on the blue button that
says “Personalize.” Adjust the mileage and the fuel price, then prompt students to compare their results
to those produced by the online tool.
Present page 9 of the flipchart. Provide students with a link to the article or present the resource
Comparison of Hybrid Cars vs Normal Cars, then ask students to discuss the questions shown on the
flipchart.
Ask: What are the key components of the modeling cycle?
Required for Honors Algebra 2:
Present page 10 of the flipchart Exploring Assumptions. Prompt students to design a model that can be
used to compute the combined fuel economy of a vehicle given 1) the MPG data for city driving, 2) the
percentage of time the driver travels under city conditions, 3) the MPG data for highway driving, and
4) the percentage of time the driver travels under highway conditions. Prompt students to compare
their calculations to those reported in the 2016 Fuel Economy Guide. Use the “Personalize” feature
found at fueleconomy.gov as a further point of comparison.
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to explain the elements of mathematical modeling and to use modeling to
solve problems.
Note to Teacher:
 This Sample Learning Task will take several days to complete.
 The resource Modeling in the CCSSM provides additional information about mathematical
modeling.
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 4: Apply the modeling cycle to model damped oscillation with combined functions.
Standard:
SMP:
F.BF.1b, F.BF.3 (F.IF.5, F.IF.7e)
1, 2
Present #1 of the MVP resource The Bungee Jump Simulator (The Bungee Jump Simulator Sample
Answers, The Bungee Jump Simulator Teacher Notes) and ensure students understand the context.
Ask:
 What do you notice and wonder about the graph (e.g., I notice that the graph oscillates like a
cosine function, but the amplitude gets smaller with each wave.)?
 What does this graph tell us about the situation (e.g., As the jumper moves a horizontal distance
along the bungee path, the vertical distance decreases with each bounce.)?
Provide time for students to work in pairs or small groups to re-produce the graphs using technology
and to determine the equation of the graph provided. To encourage perseverance, inform students that
they are not finished with the task until they can produce the exact graph and explain what functions
they used to fit the details of the graph. Ask:
 What would the graph look like if the oscillations of the bungee jumper kept returning to the
same initial height and distance from the ground as the previous oscillation (e.g., The graph
would look like a cosine curve.)?
 What do you notice about the maximum points on each oscillation (e.g., They seem to be
decreasing like an exponential curve.)?
 How does the “distance from midline” data help you understand the functions used (e.g., This
data helps me determine the amplitude of the cosine curve and rate of decay of the exponential
curve.)?
 What parameter of the equation is impacted by the data in the “distance from midline” column
(i.e., The amplitude of the cosine equation.)?
If some student groups struggle to recognize the exponential pattern in the graph, consider prompting
them to complete #2 first to activate prior knowledge about exponential functions.
As you monitor student progress, select and sequence student work and then facilitate a whole group
discussion focusing on how students decided to combine functions to model the situation. The graph in
the flipchart Bungee Jump Graph may help facilitate the discussion. Ensure that students can
summarize how combining two functions can be used to model unique situations and how they called
on their understanding of Standards for Mathematical Practice to deconstruct and re-construct the
information presented in the graph and table.
Provide time for pairs or small groups of students to complete #2 of The Bungee Jump Simulator.
Ask:
 Why did the engineer say that the change of temperature decays when the graph appears to be
increasing (e.g., The rate at which the temperature increases is decreasing.)?
 How does your friend’s conclusion that the points seem to be similar to those at the bottom of
the oscillations of the bungee jump help you write an equation (e.g., The bottom of the
oscillations in the bungee jump graph are reflections of those in the top of the bungee jump
oscillations, so I can use the same growth ratio and constant value.)?
 What other transformations have occurred to the parent exponential function that result in this
graph (i.e., a vertical shift)?
When students have completed the task, facilitate a class discussion summarizing strategies used.
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
As students reflect on their work, ensure that they can make explicit connections to each part of The
Modeling Cycle. Ask:
 How did you (or the writer of the problem) decide which variables were important to include in
your model?
 What assumptions did you make as you created your model?
 How did you validate your conclusions? What did you, or would you do next if your
conclusions did not make sense for the task?
Ask: How does combining functions impact the ability to model unique situations?
Additional Resources for Two-year Algebra 2:
Engage students in the Illustrative Mathematics resource Sum of Functions (Sum of Functions
Commentary) which includes sample answers) to access students’ prior knowledge from Unit 2 of
graphically performing arithmetic of functions. A recommendation in the commentary for this task is
to also ask students to subtract the functions. Extend the experience by asking students to also multiply
the two functions, allowing a stronger link to the primary instruction for SLT 4. These activities may
be completed prior to SLT 4.
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to use the modeling cycle to solve a damped oscillation problem.
Note to Teacher:
 This SLT is adapted from the MVP resource The Bungee Jump Simulator Teacher Notes.
 For the entire task, ensure that students have access to a variety of tools, including graph paper,
rulers, compasses, protractors, and graphing technology.
Additional Resources:
Dampening Functions Set (Dampening Functions Set Sample Answers)
LearnZillion Applications of Combining Functions
LearnZillion Combine Standard Function Types
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 5: Apply the modeling cycle to model a minimal time problem.
Standard:
F.IF.4, F.IF.5, F.IF.7
SMP:
1, 4
Use the flipchart Taco Cart to present the Dan Meyer Three-Act Task. For an overview of this type of
task, refer to the resource Three Acts Explained. Click on the image on page 1 of the flipchart to open
and play the Act 1 video. Provide time for students to process the problem and formulate a
mathematical question, and record their questions on page 2. The questions will probably be split
between finding the shorter distance and finding the shorter time for the two suggested paths, questions
that are closely related. If some students wonder about other paths, explain that they will have an
opportunity to explore those later in the activity. After students have had ample time to consider and
debate the suggested questions, turn to page 3 and ask: Which path (Ben’s or mine) will get us to the
taco cart in the least time? Prompt students to make a guess and discuss their reasoning, and follow up
by recording a tally of their votes.
Turn to Act 2 on page 4 of Taco Cart and ask: What do you need to know to answer the question (e.g.,
I need to know how far it is from the boys to the taco cart and how far it is from the boys to the road. I
need to know how fast the boys can walk in the sand or on the road.)? Prompt students to discuss their
thoughts, and record key ideas. Turn to page 5 to display two distances in feet and to page 6 to display
the rates at which the boys can walk on the road and in the sand in feet per second. Provide time for
students to work in groups of 3 or 4 to use the given information to answer the question, and offer a
variety of tools such as graph paper, rulers, compasses, protractors, and technology. Ask:
 How can you find the straight distance through the sand (i.e., I can use the Pythagorean
Theorem to find the hypotenuse of the triangle, knowing the two legs.)?
 How can you use the distances and rates to determine the times (i.e., I can divide the distance
by the rate to get the time in seconds.)?
As you circulate, identify 2 or 3 groups to present their solutions to the class, looking for variety in
methods and solutions.
Click on the image on page 7 of Taco Cart to open and play the Act 3 video, which will show which
boy gets to the taco cart first and how long it took each of them. After watching the video, provide time
for students to return to their solutions and compare them to the actual answer and rework their
calculations as necessary. Ask:
 How do your units relate to the units given in the video (e.g., My answer is in seconds, but the
video gives the answer in minutes, seconds, and hundredths of a second, so I need to make a
conversion.)?
 How close does your solution have to be to the actual time shown in the video for you to be
convinced that your computations were accurate (e.g., I would expect the solutions to not be
exactly the same, because it would be hard for the two boys to walk at exactly the given rates,
and it would be hard to measure those rates exactly and to measure the distances exactly.)?
 What factors might impact the boys’ walk that were not taken into account in your calculations
(e.g., wind, variations in the sand, the tide coming in)?
A sample solution is provided on page 16 of the flipchart for the benefit of the teacher only.
Click on the image on page 8 of Taco Cart to open and play the Sequel Act 1 video. Provide time for
students to formulate a new mathematical question, and record their ideas on page 9. Turn to page 10
and ask: Can you find another path made up of two line segments as shown in the video that results in
a shorter time? Consider giving students Taco Cart Still Shot, so they can make their own guesses, and
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
record their thoughts on the flipchart. Turn to page 12 and ask: Is there anything new you need to know
to answer the question? Turn to pages 13 and 14 to review the information that was already given.
As students apply the modeling process to identify the variables, formulate a function model, analyze
the relevant characteristics of their model, interpret their results, and validate their conclusions, ask:
 Can you sketch a graph of your function model before you find its rule (e.g., I could try several
different locations on the road that the walker should walk towards and compute or estimate
values for the time using each of those paths.)?
 Are there any limitations on the values of your variables (e.g., Distances and times have to be
positive. The total time should be reasonably close to the two values for total time found in the
original task.)?
 How does knowing reasonable domain and range values for your model function in the
problem context help you find key features of your graph (e.g., Knowing reasonable domain
and range values helps me determine a viewing window that will show the key features of the
graph on my calculator.)?
As you circulate, identify 2 or 3 groups to present their solutions to the class, looking for variety in
methods and solutions. Ensure that students have adequate time to interact with the problem before
discussing solutions.
Click on the image on page 15 of Taco Cart to open and play the Sequel Act 3 video, which will show
the path that takes the least time and how long it takes. After watching the video, provide time for
students to return to their solutions and compare them to the actual answer and rework their
calculations as necessary. A sample solution is provided on pages 17 and 18 for the benefit of the
teacher only.
As students reflect on their work with this task, ensure that they can make explicit connections to each
part of The Modeling Cycle. Ask:
 How did you decide which variables were important to include in your model?
 What assumptions did you make as you created your model?
 How did you validate your conclusions? What did you, or would you do next if your
conclusions did not make sense for the task?
Ask: How can you use the modeling cycle to find the minimal time necessary to walk from a spot on
the beach to a taco cart on the road?
Additional Resources for Two-year Algebra 2:
Extend the Taco Cart Task by asking students to investigate whether there is a location for the taco cart
that ever results in Ben winning the race.
Required for Honors Algebra 2:
Extend the Taco Cart Task by challenging students to find a path that results in an even shorter time
than the two-segment path found in the sequel. Alternatively, use the flipchart Moving the Taco Cart to
present another sequel to the Taco Cart Task.
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to use the modeling cycle to solve a minimal time problem.
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
Note to Teacher:
 The Taco Cart Task will take approximately 2 days to complete.
 For the entire task, ensure that students have access to a variety of tools, including graph paper,
rulers, compasses, protractors, and graphing technology.
 The initial task provides students with a very concrete task that only uses arithmetic. Be sure to
present the sequel as well, and ensure that students understand that finding the minimum value
of a radical function that models the situation becomes more efficient than checking a variety
of paths by arithmetic to find the shortest time.
Additional Resources:
IM: The Shortest Route - a Schoolyard Problem
Robert Kaplinsky: How Much Shorter Are Staggered Pipe Stacks?
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 6: Apply recursive processes and the modeling cycle to real-world applications.
Standards:
SMP:
F.BF.1, F.BF.2 (F.IF.5, F.IF.3)
1, 4
Provide students with the resource Modeling Credit A (Modeling Credit Sample Answers) and ensure
that they understand the context. Prompt students to work in small groups to complete the task and to
use the resource The Modeling Cycle to guide their process. Alternatively, you may choose to provide
some groups of students Modeling Credit B which contains more scaffolding. Parallel tasks have been
created for each resource in this SLT where one version (A) is denoted with the symbol
next to the
title and a more scaffolded version (B) is denoted with a  in the title. Ask:
 Why do you think Ja’Rae decided to use the credit card and make monthly payments (e.g.,
Because he did have the entire $3,000 on December 23rd.)?
 How does the modeling cycle help you complete this task?
Facilitate a class discussion about the task making explicit connections to each part of The Modeling
Cycle. Ask:
 How did you decide which variables were important to include in your model?
 What assumptions did you make as you created your model?
 How did you validate your conclusions? What did you, or would you do next if your
conclusions did not make sense for the task?
Provide students with the resources Modeling Medication Absorption A (Modeling Medication
Absorption Sample Answers) and Modeling Triangular and Oblong Numbers A (Modeling Triangular
and Oblong Numbers Sample Answers) and allow them to choose which task they’d like to complete.
Form small groups based on student choices and prompt them to complete their task clearly
demonstrating their use of each aspect of The Modeling Cycle.
Ask: How does the use of recursive processes impact the ability to model unique situations?
Additional Resources for Two-year Algebra 2:
Engage students in multiple opportunities to work towards independently implementing The Modeling
Cycle by allowing them to choose a variety of tasks from this SLT and its additional resources. Be sure
to gradually remove scaffolds with each task and provide feedback on the use of The Modeling Cycle.
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to use the modeling cycle to solve problems.
Note to Teacher:
 This SLT may take more than one day.
 For the all tasks, ensure that students have access to a variety of tools, including graph paper,
rulers, compasses, protractors, and graphing technology.
 Modeling Medication Absorption B and Modeling Triangular and Oblong Numbers B are not
mentioned in the body of the SLT above. Consider moving students flexibly among groups to
provide more scaffolding and less scaffolding as needed. Work to move students away from
needing scaffolding to independently implement the entire Modeling Cycle.
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
Additional Resources:
Recursive Processes on the TI-83/84 Calculator
IM: Snakes on a Plane (Snakes on a Plane Commentary which includes sample answers)
IM: Susita’s Account (Susita’s Account Commentary which includes sample answers)
NCTM Illuminations: Recursive and Exponential Rules
LearnZillion: Writing a recursive formula for an exponential relationship
MCPS © 2015-2016
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
SLT 7H: Apply the modeling cycle to model motion parametrically.
Standard:
SMP:
F.BF.1
1, 2
Remind students of the Unit 3 SLT 9 task, “High Noon and Sunset Shadows,” where students tracked
the motion of a shadow along the ground when the sun was directly overhead and the motion of a
shadow along the wall when the sun was on the horizon. Consider using a flashlight to model the
situations as was done in this SLT. Ask:
 What did the independent and dependent variables represent in the functions written for these
situations (e.g., The independent variable in both situations represented time. The dependent
variable in the “high noon shadow” represented horizontal distance from the center and the
dependent variable in the “sunset shadow” represented vertical distance from the center.)?
 Do either of these functions completely represent the location of the rider on the Ferris wheel
(e.g., No. Each represents only the vertical or horizontal distance of the shadow from the
center, not the location of the rider.)?
Tell students that today’s lesson will help them begin to think about how to model this type of
simultaneous horizontal and vertical motion as a function of time. Ask students to use their finger to
slowly draw the first letter of their name in the air. Use a timer to record how long students take to
trace their letter. It may be helpful to place students in groups of three and ask one student to write a
letter in the air as one partner tracks horizontal motion and the other tracks vertical motion. Or,
consider having one student demonstrate the process for the class so that all students can observe the
position of their finger as the time passes. Half of the class should track horizontal motion and the
other half track vertical motion. Ask students to estimate:
 Given the starting point of your finger as the origin, what is the horizontal position of your
finger after 2 seconds? 5 seconds (e.g., After 2 seconds, my finger has moved 1 inch
horizontally.)?
 Given the starting point of your finger as the origin, what is the vertical position of your finger
after 2 seconds? 5 seconds (e.g., After 2 seconds, my finger has not moved vertically.)?
Ask students to create tables of their horizontal positions and vertical positions dependent on time.
Present the resource High Noon and Sunset Shadows Combined (High Noon and Sunset Shadows
Combined Sample Answers, High Noon and Sunset Shadows Combined Teacher Notes) and ask
students to compare the tables they created to those presented on page 1 of this resource. Allow
students time to make sense of the situation and data presented in the tables and to determine how
these tables are similar and different from the ones they created when tracing their own letters. Ask
students to complete #1–3 of the resource. Facilitate discussion about student responses to these
questions, ensuring that students understand that these tables describe the simultaneous horizontal and
vertical positions of the picture dependent on time. Ensure that students write a prediction about the
letter formed in response to #3, then prompt them to complete #4. Ask:
 Are there times when the hand is moving more slowly or more rapidly in either the horizontal
or vertical direction (e.g., Yes. There is a time when the hand showing vertical motion moves
rapidly down, but the hand showing horizontal motion does not move at all.)?
 Do you think this letter is cursive or print? Why (e.g., The letter should be cursive so that the
graphs represent a continuous relation.)?
 Should the points we plotted in #4 be connected with straight lines or curves (e.g., If the letter
is cursive, we would expect it to smoothly curve, rather than being a series of line segments.)?
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Curriculum 2.0 Algebra 2: Unit 4-Topic 1, SLT 1–7
Provide time for students to draw a new letter in the grid for #5 and create the tables on a separate
sheet of paper that describe the horizontal and vertical motion in terms of time. Prompt students to
switch tables with a partner and try to determine the letter described by their partner’s tables.
Encourage students to attempt to determine their partner’s letter by envisioning the motion, but provide
graph paper to students who may still need to plot points to visualize the drawing.
Prompt students to complete #6 of High Noon and Sunset Shadows Combined which returns to the
context of the person on the Ferris wheel. Listen for students to transfer their learning about vertical
and horizontal motion from the letter drawing to this context. Ask:
 How is this situation similar to and different from the letter drawing (e.g., It is similar because
we are considering simultaneous horizontal and vertical motion in terms of time. It is different
because now we are thinking about trig functions to describe each motion.)?
 How could you describe the position of the rider at 5 seconds (e.g., (5, 0) horizontally and
(5,55) vertically)?
Select one or two student-generated tables to display. Ask students to predict what the figure looks like
without actually plotting points. Ask:
 On what intervals is the horizontal motion, shadow on the ground, moving slowly right (e.g., It
appears to moving slowly to the right on the time intervals [10,12.5] and [17.5,20]. )?
 On what interval is the vertical motion, shadow on the wall, moving slowly down (e.g., It
appears to be slowly moving down on the interval [5,7.5].)?
Continue to examine small intervals of time until students can predict the actual shape of the twodimensional graph.
Ask students to read #7 and note that there is a new vocabulary term in the task. Ask: Based on our
activities in this lesson, what does the term parametric equations mean (e.g., Parametric equations are
a set of equations that define a curve in terms of a separate parameter, such as time.)? Prompt students
to complete #7 of the resource based on the class discussion. Provide graph paper to students still
trying to interpret the shape of the two-dimensional graph. Facilitate a class discussion summarizing
the characteristics of the Ferris wheel and its motion.
Ask: How does using a separate parameter, such as time, allow us to gain more information about the
situation being modeled?
Check for Student Understanding: Use formative assessment processes to determine the extent to
which each student is able to use the modeling cycle and parametric equations to model situations.
Note to Teacher:
 This SLT is adapted from the MVP resource High Noon and Sunset Shadows Combined
Teacher Notes.
 This SLT may take more than one day.
Additional Resources:
High Noon and Sunset Shadows Combined Ready, Set (Sample Answers)
MVP: Parametrically-Defined Curves (Sample Answers, Teacher Notes)
MVP: Parametrically-Defined Curves Ready,Set (Parametrically-Defined Curves RS Sample Answers)
Introducing Parametric Equations, Wisconsin Math Council (includes real-world videos)
Graphing Parametric Equations on the TI-84
Graphing Parametric Equations on the TI-Nspire
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