Lesson Title: Introduction to Linear Combination
Date: _____________ Teacher(s): ____________________
Course: Common Core Mathematics 8
Start/end times: ____________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which Mathematical
Practices do you expect students to engage in during the lesson?
8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by
graphing the equations. Solve simple cases by inspection. For example, 3x  2y  5 and
3x  2y  6 have no solutions because 3x  2y cannot simultaneously be 5 and 6.
MP1:
MP2:
MP3:
MP4:
MP5:
MP6:
MP7:
MP8:
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.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Have students identify strategies discovered so far for
finding a solution to a system of equations. When is it
best to use each of the strategies?
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow? List the questions.
What does linear combination allow us to find? Why
might linear combination be useful?
(Look for evidence of MP7 and MP8.)
(Look for evidence of MP3.)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
Note: This lesson was inspired from an activity available in the PBS Mathline Lesson “The Busing Problem.” This
lesson is designed to help students understand conceptually why the linear combination method works. The lesson
following this lesson should focus on using linear combination to solve a system.
1. Discuss the lesson launch. Tell students that today we will be discovering another method for solving systems of
equations.
2. Assign students to pairs. Each pair will be assigned a system of equations. First, the pair should graph the
system on the student resource sheet. The pair should identify the solution to the system on the graph and label
the point. Each person should be completing work on his/her own paper. (Look for evidence of MP1, MP4,
MP5, MP6, MP7, and MP8.)
3. Now, tell students to add the two equations together to get a third equation. Have students simplify the equation
and graph the new line on the same coordinate grid using a black pen or colored pencil. It may be helpful to
have an additional system ready as a sample. Ask students what they notice about this new line. Sample
response: The new line also passes through the point of intersection, so it has the same solution. Once students
identify this, students should write this finding below their graph, i.e. When I add two equations together, the
new equation formed will have the same solution. (Look for evidence of MP1, MP2, MP6, MP7, and MP8.)
4. Pairs will now travel to different stations around the room. At each station, students will perform the specified
operation(s) to the original system to produce a new line and then draw the new line on the same coordinate grid
using the designated color pencil. Pairs should write their findings below the coordinate grid. (Look for
evidence of MP4, MP5, MP6, MP7, and MP8.)
5. Once pairs have traveled to each station, reform groups so that students can compare results with students that
had different systems. Were the findings the same? Did it matter that they started with different systems? What
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Introduction to Linear Combination
Course: Common Core Mathematics 8
Date: _____________ Teacher(s): ____________________
Start/end times: ____________
findings did the groups have in common? (Look for evidence of MP2, MP3, MP7, and MP8.)
6. Have class discuss findings and identify what operations are possible to create new equations that will still have
the same solution to the system of equations. This strategy is known as linear combination. Why might this
strategy be useful? (Look for evidence of MP1, MP2, MP3, MP4, MP7, and MP8.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened (and conceptual) understanding.
Students will be able to use a linear combination strategy to produce a new equation that has the same solution as the
original solution. Students will be able to understand that this strategy can be used to solve a system of equations
without graphing.
The teacher will be able to measure student understanding by monitoring group work at stations as well as class
discussion.
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Key vocabulary: solution, intersection, linear combination
Connections: Students will need to connect to their prior understanding of the solution to systems of equation
represented as the intersection of the graphs.
Common Mistakes: Students may think that only the particular linear combinations examined will produce a
solution but not able to understand the key concepts of adding/subtracting equations or multiplying by any scalar
multiple will result in a viable solution.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Student resource sheet (large coordinate grid)
Colored Pencils
One system per pair
Color Stations
Graphing calculators (optional)
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
How well do students understand how linear combinations can be used as a strategy for solving systems of
equations?
Will students be able to apply this understanding to solve systems using linear combination in the next lesson?
How will I differentiate upcoming lessons to meet the needs of the various learners in my class?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product
under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.