STEM-Centric Unit
Systems of Equations
Author: Hannah Knisely Kent Island High School, Queen Anne’s County Public Schools
Background Information
Subject:
Identify the course the unit will be implemented in.
Grade Band:
Identify the appropriate grade band for the lesson.
Duration:
Identify the time frame for the unit.
Overview:
Provide a concise summary of what students will
learn in the lesson. It explains the unit’s focus,
connection to content, and real world connection.
Background Information:
Identify information or resources that will help
teachers understand and facilitate the challenge.
STEM Specialist Connection:
Describe how a STEM Specialist may be used to
enhance the learning experience. STEM Specialist
may be found at http://www.thestemnet.com/
Enduring Understanding:
Identify discrete facts or skills to focus on larger
concepts, principles, or processes. They are
transferable - applicable to new situations within or
beyond the subject.
Algebra I
9-12
90 Minutes
In this lesson, students will build on experiences with solving systems of equations
from middle school to focus on justification of the methods used and the real-world
application of equations. The lesson begins with a review of systems of equations.
Students then solve for unknowns and justify the process employed for solution
development. They will assess the most efficient method for solving systems. They
will then apply their knowledge to solve real-world problems.
Teachers should have a strong background in solving systems of equations through
substitution, elimination, and/or graphing for an intersection point. At Khan Academy
you can view videos and practice simple linear systems. The solution to systems of
equations can be infinite (meaning it’s the same function), a single set of ordered
pairs, multiple sets of ordered pairs, or no solution at all (The functions never cross).
Systems of equations have numerous real-world applications. Throughout the lesson,
students will be responsible for interpreting situations, solving for two unknowns, and
making sense of the answers through analysis of the results.
STEM Specialists can be used to engage students in hands-on learning experiences
that allow them to use systems of equations to develop solutions to unknowns.
Specialist can help students identify careers that regularly use systems of equations.


Mathematical models are used to develop solutions to real-world problems.
Systems of equations can be used to find multiple unknowns (significant
events, break-even points, etc.)
Page 1 of 18
STEM-Centric Unit
Systems of Equations
Background Information
Essential Questions:



How are systems of equations applied in the real-world?
How does one determine the most efficient method to solve systems?
How are the symbolic, numeric, graphic, and verbal representations of
functions and equations related?
Students will be able to:
Student Outcomes:
1. Solve systems of equations using various methods (substitution, elimination,
Identify the transferable knowledge and skills that
and/or graphing) and identify the most efficient method.
students should understand and be able to do when
the lesson is completed. Outcomes must align with
2. Identify the mathematical property used at each step in the solution process as
but not limited to Maryland State Curriculum and/or
a means of justifying a step.
national standards.
3. Analyze how systems of equations are used in the workforce.
Audience:
☒Peers

Students
will
solve
several
real-world
problems
involving
☒Experts /
Product, Process, Action, Performance,
systems of equations.
Practitioners
etc.:
 Students will create their own problem, implementing a
Identify what students will produce to
☒Teacher(s)
demonstrate that they have met the challenge,
STEM concept, which must be solved by a peer. The
☐School
learned content, and employed 21st century
student must also include an answer key indicating that an
Community
skills. Additionally, identify the audience they will
answer exists and makes sense in the context of the
present what they have produced to.
☐Online
problem.
Community
☐Other______
Common Core Algebra I Standard:
Domain: Linear and Exponential Relationships
Cluster: Solve systems of equations
Standard: A.REI.5: Prove that, given a system of two equations in two
Standards Addressed in the Unit:
variables, replacing one equation by the sum of that equation and
Identify the Maryland State Curriculum Standards
multiple of the other produces a system with the same solutions.
addressed in the unit.
Identify several open-ended questions to provoke
inquiry about the core ideas for the lesson. They are
grade-level appropriate questions that prompt
intellectual exploration of a topic.
Standard: A.REI.C.6: Solve systems of linear equations exactly and
approximately (e.g., with graphs), focusing on pairs of linear equations
in two variables.
Page 2 of 18
STEM-Centric Unit
Systems of Equations
Suggested Materials and Resources:
Identify materials needed to complete the unit. This
includes but is not limited to websites, equipment,
PowerPoints, rubrics, worksheets, and answer keys.
Background Information
Equipment:
 Computer
 Projector
 Graphing calculators (optional)
People:
STEM Specialist
Materials (rubrics, worksheets, PowerPoints, answer keys, etc.):
 Candy Cost Worksheet
 Candy Cost Worksheet (Answer Key)
 Systems of Equations PowerPoint
 Homework Assignment
 Homework Assignment (Answer Key)
Page 3 of 18
STEM-Centric Unit
Systems of Equations
Duration: 90 Minute Class
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
☒Engagement
☐Exploration
☐Explanation
☐Extension
☐Evaluation
Details
Materials:
 Computer
 Projector
 Systems of Equations PowerPoint
Preparation:
Use the PowerPoint slides 1-6 as a guide through the engage
portion of the learning experience.
Facilitation of Learning Experience:
 Ask students to pick two numbers. Have them label one number
“x” and the other “y”.
 Have students add the two numbers and write it as an equation (i.e.
𝑥 + y = #). Next, allow students to subtract the two numbers and
write it as an equation (i.e. 𝑥 − y = #). On a blank piece of paper,
students will write down their equations only (refer to PowerPoint
slide 3 for an example. Students will partner with someone and
switch papers. The classmate will need to identify what x and y
equals. Students will have to explain how they determined their
answers. Students will check their answers with their partners.
 Call on students to explain their strategy for finding two unknown
variables. Questions to prompt discussion include:
o What process did you use to solve the problem?
o Was there any other process you could have employed?
o Why did you select that process that you used?
o For students you used a “guess and check” approach – Was
Standards for
Mathematical Practice
☐Make sense of problems
and persevere in solving
them.
☒Reason abstractly and
quantitatively.
☒Construct viable arguments
and critique the reasoning
of others.
☐Model with mathematics.
☐Use appropriate tools
strategically.
☐Attend to precision.
☒Look for and make use of
structure.
☐Look for and express
regularity in repeated
reasoning.
Page 4 of 18
STEM-Centric Unit
Systems of Equations
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
Standards for
Mathematical Practice
this the most efficient process for developing a solution? Why
or why not?
Have students openly discuss the methods they used to solve the
problem. When the classroom discussion ends (or someone mentions a
method similar to elimination) show the students how the problem can be
solved using elimination (refer to PowerPoint slide 6 for an example).
☐Engagement
☐Exploration
☒Explanation
Materials:
 Computer
 Projector
 Graphing calculators (optional)
 Candy Cost Worksheet
 Candy Cost Worksheet (Answer Key)
☒Make sense of problems
and persevere in solving
them.
Preparation:
 Use PowerPoint slides 7-17 as a guide through the exploration
portion of the learning experience.
☐Construct viable arguments
and critique the reasoning
of others.
☐Extension
☐Evaluation
Facilitation of Learning Experience:
 Display PowerPoint slide 7. Explain to students that there are
different ways to solve for unknown variables using equations.
 Explain the steps for substitution, then elimination, then graphing.
 Work through the candy bag problem as a class. There is a
worksheet to go with this activity if you choose to use it. The activity
can be completed without the worksheet as well.
 Read the situation carefully, then make a table of values to
determine how many bags Mrs. Drawbaugh bought.
 Next show the students how to solve this problem using
☒Reason abstractly and
quantitatively.
☐Model with mathematics.
☐Use appropriate tools
strategically.
☐Attend to precision.
☒Look for and make use of
structure.
☐ Look for and express
Page 5 of 18
STEM-Centric Unit
Systems of Equations
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details

☐Engagement
☒Exploration
☐Explanation
Standards for
Mathematical Practice
regularity in repeated
substitution, elimination, and on a graph (this can be done using
reasoning.
their calculators).
As a class discuss which option is the best to use. Are there times
where one method is better than another? Discuss that the format in
which the equations are presented makes a difference as to which
method you would use over another.
Materials:
 Computer
 Projector
☐Make sense of problems
and persevere in solving
them.
Preparation:
 Contact the STEM Specialist in advance to engage students in
☐Reason abstractly and
quantitatively.
Page 6 of 18
STEM-Centric Unit
Systems of Equations
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
☐Extension
☐Evaluation
Details
hands-on learning experiences that demonstrate to students how
workplace professionals use equations to develop solutions to
unknown variables in the real-world. Co-plan the lesson with the
STEM Specialist. A description of the ability level of the students,
as well as some of the prior knowledge your students will be helpful
to the STEM Specialist prior to the learning experience. STEM
Specialists can be found at (www.theSTEMnet.com)
Facilitation of Learning Experience:
 It is a suggested that a STEM Specialist be used for this portion of
the lesson to help students understand and apply course content to
real world scenarios. Co-plan this learning experience with the
STEM Specialist.
 An alternate approach is to use PowerPoint slides 18-28 to
demonstrate how what they have learned is applied to the real
world. Begin by asking students, “When are you ever going to use
systems of equations to find unknowns?” Listen and discuss their
answers before showing the PowerPoint slides. Have students
come up with situations in which they believe systems of equations
would be useful for finding unknowns.
 The examples provided in the PowerPoint include:
o Business profit vs. Expenses (break-even point)
o Package Comparison—Verizon vs. AT&T (when is one
company’s prices better than another?). Feel free to expand
upon this concept as it lends itself well to other examples.
o Population Growth vs. Food Supply (When will we reach a
population level in which we can no longer physically feed
everyone? What does that mean for the world?)
Standards for
Mathematical Practice
☒Construct viable arguments
and critique the reasoning
of others.
☒Model with mathematics.
☐Use appropriate tools
strategically.
☐Attend to precision.
☐Look for and make use of
structure.
☐ Look for and express
regularity in repeated
reasoning.
Page 7 of 18
STEM-Centric Unit
Systems of Equations
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
Standards for
Mathematical Practice
o A secondary population growth vs. food supply chart that
includes death rate, birth rate, resources, population, etc.
o Economics (Supply and Demand)
o A crane’s maximum capacity level (When will the cord snap
and/or the machine tip over from the weight)
o Building bridges (When will the two pieces intersect?)
o And asteroid and the Earth. (Will the asteroid hit the earth?)
Perhaps you can discuss the meteor belt and the likelihood
of an asteroid hitting the Earth at the exact right moment in
its orbital path. This would be a great time to incorporate a
little bit of probability as well.
☐Engagement
☐Exploration
☐Explanation
☐Extension
☒Evaluation
Materials:
 Homework Worksheet
 Homework Worksheet (Answer Key)
☒Make sense of problems
and persevere in solving
them.
Preparation:
Complete each of the homework problems in advance in order to
determine where the students may have questions as well as to
determine how one might easily use one of the three methods for
solving the systems of equations.
☒Reason abstractly and
quantitatively.
☐Construct viable arguments
and critique the reasoning
of others.
Facilitation of Learning Experience:
☐Model with mathematics.
Allow students time to work independently. If they really struggle and
are unable to set problems up or solve for the unknown variables than
☐Use appropriate tools
guidance should be given. Do not solve any of the problems for the
strategically.
students. This is the time they should think for themselves and
determine the best course of action to take in order to properly solve for
Page 8 of 18
STEM-Centric Unit
Systems of Equations
Learning Experience
5E Component
Identify the 5E component
addressed for the learning
experience. The 5E model
is not linear.
Details
the unknown variables.
Standards for
Mathematical Practice
☒Attend to precision.
☒Look for and make use of
structure.
☒ Look for and express
regularity in repeated
reasoning.
Page 9 of 18
STEM-Centric Unit
Systems of Equations
Interventions/Enrichments
Identify interventions and enrichments for
diverse learners.
Supporting Information
Struggling Learners
Struggling learners should be provided with guided notes in order to better
understand the application of each of the methods for solving systems of
equations. If they have the steps previously listed for them they can focus their
attention on watching the teacher solve an example problem. Additionally, the
STEM specialist can work with these students to provoke answers as to where
one might use this concept in the real world (specifically STEM careers). Some
additional example problems may help struggling learners better see how to
complete each method.
English Language Learners
English Language Learners would benefit from the accommodations provided
to struggling learners. Additionally, a word bank with definitions of vocabulary
may be helpful to better understand the three different methods for solving
systems of equations. Depending on the regions in which your English
Language Learners are from, perhaps examples specifically from their native
country may help them connect to the information better.
Gifted and Talented
Gifted and Talented students should be pushed to further understand the
concepts. Have them research a specific STEM career, that interests them,
and investigate how an individual in that position would use systems of
equations to solve for unknowns. Rather than having them simply solve more
problems, or solely focus on the math, allow them to branch out and
understand the situation as a whole (cross curricular).
Page 10 of 18
Candy Costs
Name: ________________________
Mrs. Drawbaugh decided to purchase
candy for her whole class as a treat.
She bought Smarties and Dum-Dum
lollipops as “brain food” for their next
exam. Each bag of Smarties cost
$7.00 (including tax). The bag of
Dum-Dum lollipops cost $8.50
(including tax). She ended up
spending $60.50 on her purchase of 8 items.
1. Using the information above, complete the following table:
Number of Smarties Bags
0
1
2
3
4
5
6
7
8
Number of Dum-Dum lollipops Bags
2. Circle the row that has a total cost of $60.50.
3. How many bags of Smarties did Mrs. Drawbaugh buy?
4. How many bags of Dum-Dum lollipops did Mrs. Drawbaugh buy?
Total Cost for 8 Items ($)
Candy Costs
Name: ________________________
5. Define your variables.
6. Write a system of equations to model the situation.
7. How many bags of Smarties did Mrs. Drawbaugh buy?
8. How many bags of Dum-Dum lollipops did Mrs. Drawbaugh buy?
Candy Costs (Answer Key)
Mrs. Drawbaugh decided to purchase
candy for her whole class as a treat.
She bought Smarties and Dum-Dum
lollipops as “brain food” for their next
exam. Each bag of Smarties cost
$7.00 (including tax). The bag of
Dum-Dum lollipops cost $8.50
(including tax). She ended up
spending $60.50 on her purchase of 8 items.
1. Using the information above, complete the following table:
Number of Smarties Bags
0
1
2
3
4
5
6
7
8
Number of Dum-Dum lollipops Bags
8
7
6
5
4
3
2
1
0
Total Cost for 8 Items ($)
$68.00
$66.50
$65.00
$63.50
$62.00
$60.50
$59.00
$57.50
$56.00
2. Circle the row that has a total cost of $60.50.
3. How many bags of Smarties did Mrs. Drawbaugh buy?
5
4. How many bags of Dum-Dum lollipops did Mrs. Drawbaugh buy?
3
Candy Costs (Answer Key)
5. Define your variables.
x = number of Smarties bags
y = number of Dum-Dum lollipop bags
6. Write a system of equations to model the situation.
Items: 𝑥 + 𝑦 = 8
Cost: 7.00𝑥 + 8.50𝑦 = 60.50
7. How many bags of Smarties did Mrs. Drawbaugh buy?
Finding the number of Smarties bags means I should eliminate y since that is the variable that
defines Dum-Dum lollipop bags.
𝑥 + 𝑦 = 8 (multiply this equation by -8.50 to eliminate y)
7.00𝑥 + 8.50𝑦 = 60.50 (nothing needs to change here)
−8.50𝑥 − 8.50𝑦 = −68
+
7.00𝑥 + 8.50𝑦 = 60.50
−1.50𝑥
−1.50
=
−7.50
−1.50
𝑥=5
8. How many bags of Dum-Dum lollipops did Mrs. Drawbaugh buy?
Finding the number of Dum-Dum lollipop bags means I should eliminate x since that is the
variable that defines Smarties bags.
𝑥 + 𝑦 = 8 (multiply this equation by -7.00 to eliminate x)
7.00𝑥 + 8.50𝑦 = 60.50 (nothing needs to change here)
−7.00𝑥 − 7.00𝑦 = −56
+
7.00𝑥 + 8.50𝑦 = 60.50
1.50𝑦
1.50
=
4.50
1.50
𝑦=3
Systems of Equations Homework
Read the follow situations carefully. You may use substitution, elimination, or graphing
to answer the questions below.
1. Beth is deciding between two phone plans. Plan A charges $15 per month plus 10 cents per
minute she talks on the phone. Plan B charges $20 per month but only charges 5 cents per
minute she talks on the phone.
a. Write a system of equations to represent the monthly cost of each plan.
b. Solve the system using any method you prefer.
c. How many minutes per month do the phone plans cost the same amount?
2. Austin and Nicole are playing a game where they toss a dart at a game board that is hanging
on the wall. The points earned from a toss depends on where the dart lands. The center area
is worth more point than the surrounding areas. Each player tosses 12 darts.
a. Austin earned a total of 66 points with 6 darts landing in each area. Nicole earned a
total of 56 points with 4 darts landing in the center area, and 8 darts landing in the
surrounding area. Write a system of equations that represents the number of darts
each player tossed into each section. Use x for the inner circle, and y for the outer
circle.
b. How many points is the inner circle worth? How many points is the outer circle worth?
c. If a player gets 10 darts in the inner circle and 2 in the outer circle the total score is
doubled. How many points would the player earn if he or she gets exactly 10 darts in
the center?
3. Find the points of intersection between the line y = x – 4 and the circle x2 + y2 = 4.
4. Sarah received presents and cards from friends over the holiday season. Every present came
with one card and none of her friends sent her more than one card. Less than 10 of her
friends sent only a card. Describe this situation using inequalities.
5. The county is planning to build a new high school. To furnish it, the county is willing to spend
$10,400 on tables and chairs. The tables they’ve picked out cost $200 and the chairs cost $40.
Each table can fit 8 chairs around it. How many tables and chairs will the county purchase?
6. An airplane is traveling along the line 𝑥 − 𝑦 = −1 when it sees another airplane traveling along
the line 5𝑥 + 3𝑦 = 19. If they continue along the same lines, at what point will their flight
paths cross?
What does this point represent in the context of the problem?
Now it’s your turn
Think of a real-world situation in which you may use systems of equations. Create your own problem
from this thought. Do some research on an event and once you feel you understand the situation
enough create your own word problem below. You will need to write the situation, determine the
system of equations, and work the solution out yourself—solve the problem.
Systems of Equations Homework Answer Key
Read the follow situations carefully. You may use substitution, elimination, or graphing
to answer the questions below.
1. Beth is deciding between two phone plans. Plan A charges $15 per month plus 10 cents per
minute she talks on the phone. Plan B charges $20 per month but only charges 5 cents per
minute she talks on the phone.
a. Write a system of equations to represent the monthly cost of each plan.
𝑦 = 0.10𝑥 + 15
𝑦 = 0.05𝑥 + 20
b. Solve the system using any method you prefer.
Work will vary
c. How many minutes per month do the phone plans cost the same amount?
100 minutes
2. Austin and Nicole are playing a game where they toss a dart at a game board that is hanging
on the wall. The points earned from a toss depends on where the dart lands. The center area
is worth more point than the surrounding areas. Each player tosses 12 darts.
a. Austin earned a total of 66 points with 6 darts landing in each area. Nicole earned a
total of 56 points with 4 darts landing in the center area, and 8 darts landing in the
surrounding area. Write a system of equations that represents the number of darts
each player tossed into each section. Use x for the inner circle, and y for the outer
circle.
6𝑥 + 6𝑦 = 66
4𝑥 + 8𝑦 = 56
b. How many points is the inner circle worth? How many points is the outer circle worth?
𝑖𝑛𝑛𝑒𝑟 𝑐𝑖𝑟𝑐𝑙𝑒 = 8 𝑝𝑜𝑖𝑛𝑡𝑠
𝑜𝑢𝑡𝑒𝑟 𝑐𝑖𝑟𝑐𝑙𝑒 = 3 𝑝𝑜𝑖𝑛𝑡𝑠
c. If a player gets 10 darts in the inner circle and 2 in the outer circle the total score is
doubled. How many points would the player earn if he or she gets exactly 10 darts in
the center?
172 points
3. Find the points of intersection between the line y = x – 4 and the circle x2 + y2 = 4.
(0, -2) & (2, 0)
4. Sarah received presents and cards from friends over the holiday season. Every present came
with one card and none of her friends sent her more than one card. Less than 10 of her
friends sent only a card. Describe this situation using inequalities.
Let p be the number of presents and c be the number of cards. 𝑝 ≤ 𝑐 and 𝑐 − 𝑝 < 10
5. The county is planning to build a new high school. To furnish it, the county is willing to spend
$10,400 on tables and chairs. The tables they’ve picked out cost $200 and the chairs cost $40.
Each table can fit 8 chairs around it. How many tables and chairs will the county purchase?
20 tables and 160 chairs
6. An airplane is traveling along the line 𝑥 − 𝑦 = −1 when it sees another airplane traveling along
the line 5𝑥 + 3𝑦 = 19. If they continue along the same lines, at what point will their flight
paths cross?
(2,3)
What does this point represent in the context of the problem?
The crash point between the planes. This coordinate point represents a geographical point on
a map (e.g. 2 east and 3 north)
Now it’s your turn
Think of a real-world situation in which you may use systems of equations. Create your own problem
from this thought. Do some research on an event and once you feel you understand the situation
enough create your own word problem below. You will need to write the situation, determine the
system of equations, and work the solution out yourself—solve the problem.
Answers will vary