Drafting Solutions
Multistep Equations in Context
No one would argue that
presentation doesn’t
matter. I can think of
thousands of examples
where I choose higher
quality presentation over
lesser… in the things I
buy, listen to, look at,
hire, etc. In fact, sometimes I even find myself
guilty of choosing
presentation over quality
of content: like that gorgeous cookie I ate yesterday that tasted awful!
This lesson is part of a
series of problems I created that are focused
around algebraic applications and the use of new
algebraic tools (tables,
equations, and graphs)
to solve problems in context. But underneath the
rich content is an important lesson about
presentation: a solution
to a problem illustrates,
generalizes, communicates, and verifies
the results.
An answer is
just a number.
Essential Questions

What is the difference between an answer and a solution?

How important is presentation?
Objectives & the Common Core Standards
N-Q (1-2): Reason quantitatively and use units to solve problems.
 Students will choose and interpret scale and origin in graphs.
 Students will use units to interpret solutions in context.
A-REI (1): Understand solving equations as a process of reasoning and explain the reasoning.
 Students will explain each step in solving a simple equation.
 Students will construct a viable argument to justify a solution method.
8.EE (7) and A-REI (3): Solve linear equations in one variable.
 Students will solve multistep equations in one variable, including equations with variables on
both sides.
 Students will analyze and solve pairs of simultaneous linear equations.
8.EE (8) and A-REI (10-11): Represent and solve equations graphically.
 Students will represent solutions to equations as a set of data plotted on a coordinate system.
 Students will understand that solutions to a system of two linear equations corresponds to
points of intersection of their graphs.
F-IF (4-6)*: Interpret functions that arise in applications in terms of the context.
 Students will calculate and interpret key features of graphs and tables (intercepts, rate of
change, end behavior, etc.)
8.F (1-2) and F-IF (7-9)*: Analyze functions using different representations.
 Students will represent functions symbolically, graphically, and numerically (in data sets).
8.F (4) and F-BF (1a)*: Write a function that describes a relationship between two quantities.
 Students will generate tables from context and determine an explicit expression to generalize
the function’s behavior.
S-ID (7)*: Interpret linear models.
 Students will interpret the rate of change and vertical intercept of a linear model in context.
Teaching Notes
This lesson is the cornerstone to a unit that is easily the most valuable bit of algebra that I can offer my students. There is no replacement to lessons learned here
about understanding functions in context through multiple representations. If
you are just beginning to make a switch to standards based grading, these lessons
will serve as a gentle easement into problem solving as a larger entity. No repetitive
problem sets here, not today. This unit will serve as a map through Bloom’s Taxonomy,
leading to enhanced skills in analysis, synthesis, and evaluation.
© 2011, Emily McGary Allman for distribution on www.TeachersPayTeachers.com/Store/The-Allman-Files. Thank you for your patronage!
This lesson is used in my class towards the beginning of the term, after:
 Students are already familiar with algebraic expressions and can solve simple multi-step equations in one
variable (including equations with variables on both sides). They don’t need to be experts, but this shouldn’t
be an intro to variables on both sides!
 Students are already familiar with the nature of functions (but not necessarily function notation).
Begin the unit by posting the first essential question on the board: “What’s the difference between an answer to
a problem and a solution to a problem?” Encourage a general discussion—don’t restrict the conversation to just
math problems. Jot some student thoughts and notes on the board, but don’t worry too much about resolution.
Let your students know that you are trying to answer that question better through an exploration of several
problems over the next few days.
The Plant Problem: A scientist plants a seed and makes a few observations. Student analysts walk through a series of directed questions guiding them through the use of tables, expressions, and graphs to
represent a function and present their findings.
I find it best to work through the Plant Problem together as a class, with teacher direction,
pausing often to ask “What’s the best way to communicate that?” “What kind of scale and
axes should we use to get the best graph?” “What’s the best answer to this part?” “Could we
say that in another way?” “What does that mean?” etc.
The Car Problem: Competing offers for a car rental leaves one to wonder which deal is better. This
problem is best presented cooperatively (pairs are great) with little or no teacher interaction. I find that students are more motivated to analyze the quality of their responses
when working in pairs (vs alone). As the teacher, I find it difficult to stay out of things,
especially when they are asking questions, but I need to keep reminding myself that
the questioning and EXPERIMENTING are the keys to their learning. I encourage questions, but try to answer with another question of my own: “What did you try already?” “What’s the domain
here?” “Did you see this part over here?” etc.
An important part of this problem is group critique. Choose a few graphs, tables, equations, and written solutions to display for public (anonymous) critique and talk about what makes one better than another.
Follow Up
To maximize impact, it is important that we follow every lesson with both practice and assessment. I always follow these problems with two similar exercises: one as homework and another as in-class practice. The assessment I offer is a problem that has both individual and cooperative elements. It is graded with a hybrid system
that includes 0-4 standards-based rubric and a 10 point checklist. Both my practice problems and assessment are
available for purchase on Teachers Pay Teachers.
Practice Set: “Performance Task Practice: Multistep Equations for Algebra 1”
http://www.teacherspayteachers.com/Product/Performance-Task-Practice-Multistep-Equations-for-Algebra-1
Assessment: “Performance Task Assessment: Multistep Equations for Algebra 1”
http://www.teacherspayteachers.com/Product/Performance-Task-Assessment-Multistep-Equations-for-Algebra-1
© 2011, Emily McGary Allman for distribution on www.TeachersPayTeachers.com/Store/The-Allman-Files. Thank you for your patronage!
The Plant Problem
Name _________________________________
Intro to Multistep Equations in Context
Professor Botano is gathering data on the growth rate of a certain new
hybrid seed. He spilled coffee on his clipboard and destroyed most of
the data, but he DID remember that the seed had been growing at a constant rate throughout his observations. Help him reconstruct the data.
 Figure out the missing values for Professor Botano’s table below:
# of days since seed
was planted
Height of seedling
(in inches)
0
2
3
4
6
7
12
8
 What is the plant’s daily growth rate? ___________________
 What is a possible explanation for the number in the height
spot on day 0?
 Write a function for the height of the seedling
in terms of days (use h for height and d for days): ______________________________
 Determine and explain the domain and range of your function.
© 2011, Emily McGary Allman for distribution on www.TeachersPayTeachers.com/Store/The-Allman-Files. Thank you for your patronage!
 Think, show, and interpret!
How tall was the plant on day 1?
When will the plant be 20 inches tall?
 Use this grid to create your best possible line graph that shows the height of the plant in
terms of days.
Professor Botano’s Hybrid Seed Study: an Analysis of Height over Time
d
h
0
2
3
4
6
8
© 2011, Emily McGary Allman for distribution on www.TeachersPayTeachers.com/Store/The-Allman-Files. Thank you for your patronage!
7
12
Teacher’s Key
The CAR Problem
Name _________________________________
Multistep Equations in Context
Suppose the Allmans want to rent a convertible for the day.
They have a choice of two rental companies:


A one-day rental at Nifty Car Rental costs $30 plus 60
cents per mile.
A one-day rental at Shazam Car Rental costs $55 but
only charges 35 cents per mile.
 Make a good data table:
# of miles driven
Nifty
Total Cost ($)
Shazam
Total Cost ($)
 Write a function for each car rental company that expresses the total cost in terms of
the number of miles driven. (Use C for cost and m for miles.)
Nifty Car Rental
Shazam Car Rental
 If the Allmans drive 225 miles, which company would be a better deal? Explain.
 If the Allmans only have $80 to spend, which company would be a better deal? Explain.
© 2011, Emily McGary Allman for distribution on www.TeachersPayTeachers.com/Store/The-Allman-Files. Thank you for your patronage!
 Use your functions and tables to make your best possible comparison graph
that represents the costs
for both companies.
Nifty vs Shazam Car Rental Comparisons
 At what number of miles will the two companies cost the same? ___________
Circle the place on the graph that verifies this.
Then use your functions to prove your
solution with an algebraic method.
 Which car rental company should the Allmans choose and why?
© 2011, Emily McGary Allman for distribution on www.TeachersPayTeachers.com/Store/The-Allman-Files. Thank you for your patronage!
Teacher’s Key



I appreciate your patronage!
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Emily @ “The Allman Files”
http://www.teacherspayteachers.com/Store/The-Allman-Files
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http://coremath912.blogspot.com/
email: coremath912@gmail.com