Personalized Learning – Old Skool Classics
(90-minute Block – Algebra 1 Options)
Subject: Algebra 1
A.
Big Ideas for the Unit
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B.
Solving systems of equations using a variety of methods (graphing, substitution,
combination, and Cramer’s Rule)
Graphing systems of equations and inequalities
Using matrices
Key Vocabulary for the Unit
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C.
Lesson Title: Solving Linear Systems by Substitution
System of two linear equations
Consistent
Inconsistent
Substitution Method
Combination Method
Matrix
Cramer’s Rule
System of linear inequalities
Common Core Standards:

Unit 1 - Relationships between quantities and reasoning with equations: Reason
quantitatively and use units to solve problems; interpret the structure of
expressions; create equations that describe numbers and relationships;
understand solving equations as a process of reasoning and explain the
reasoning; solve equations and inequalities in one variable.
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Unit 2 - Linear and Exponential Relationships: Solve systems of equations;
represent and solve equations and inequalities graphically; interpret functions
that arise in applications in terms of a context; build a function that models
relationships between two quantities.
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Mathematical Practice Standards: 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.
D.
Learning goals for lesson:
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Students will know how to solve a system of linear equations algebraically by using
substitution.
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Students will be able to analyze and model real life situations.
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Essential question: How would you solve a linear system by substitution?
E.
Implementing the lesson
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Pre-assessment or background knowledge: Students must be able to solve multistep equations and know the concept of a solution.
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Materials and management considerations: 3 leveled worksheets (basic, average,
and advanced).
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Personalization / Tiering options:
1. What will be tiered – content, process, or product? process
2. How will the lesson be tiered – by readiness, interest, or style? readiness
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Warm-up/Review:
Students will work in groups of two. Randomly assign a student pair to address
each problem by writing their group answer on the board. Other students
should check their responses against those posted.
1. What does it mean “to solve a system of linear equations algebraically”?
2. What are the three possible scenarios and what conclusion can you reach
respect to the number of solutions to a linear system? Give a graphical
example of each scenario?
3. Decide whether (-3,5) is a solution of the linear system
2x –5y = -31; -3x + y = 14
4. Sketch a graph to solve the system: y = 2x + 3; 3x+ 2y = -1
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Direct / Whole group instruction (as needed):
Steps to Solving a Linear System by Substitution:
1. Solve one of the equations for one of its variables. Solve for the
easiest variable, e.g., the variable that has a coefficient of ± 1.
2. Substitute the expression from Step 1 into the second equation and
solve for the other variable.
3. Substitute the value from Step 2 into the revised equation from Step
1 above and solve for that variable.
4. Check the solution in each of the original equations.
Example: y - x = – 4
4x + y = 26
Equation 1
Equation 2
Step One: Use the first equation and solve for y: y = x – 4
Step Two: Substitute “x – 4” for y in Equation 2 and solve for x.
4x + y
= 26
4x + (x – 4) = 26
5x
= 30
x = 6
Step Three: To find the value of y, substitute 6 for x in the revised Equation 1
and simplify. y = x – 4; y = 2
Step Four: Check to see if (6,2) is the solution by substituting 6 for x and 2
for y in each of the original equations.
F.
Guided Practice during Class
Students will be grouped in three tiers based on math readiness: Tier 1, Tier 2
or Tier 3. Real life application problems will be shared with each group. They will
work independently as the teacher floats around to offer assistance as needed.
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Tier 1 - Basic Learners: Students will work on four basic problems and
select any one of the application problems to complete. Students will be
given an example problem for them to follow.
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Tier 2 - Grade Level Learners: Students will work on 3 problems of medium
difficulty, along with any three of the application problems.
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Tier 3 - Advanced Learners: Students will work on 1 medium difficulty
problem plus 2 advanced problems, along with four of the application
problems.
G.
Exit Ticket: (Error Analysis assessment). Give students an incorrect step by step
problem and let them figure out and/or make the appropriate corrections.
H.
Homework: Students will be given additional practice exercises from the textbook
using a “Tic-Tac-Toe” framework.
Tier 1 Worksheet - Guided Practice
Steps to Solving a Linear System by Substitution:
1. Solve one of the equations for one of its variables. Solve for the
easiest variable, e.g., the variable that has a coefficient of ± 1.
2. Substitute the expression from Step 1 into the second equation and
solve for the other variable.
3. Substitute the value from Step 2 into the revised equation from Step
1 above and solve for that variable.
4. Check the solution in each of the original equations.
Example:
2x + 5y = -5
x + 3y = 3
Equation 1
Equation 2
Step One: Use the second equation and solve for x: x = -3y +3
Step Two: Substitute “-3y +3” for x in Equation 1 and solve for y.
2x + 5y = -5
2(-3y +3) + 5y = -5
- 6y + 6 + 5y = -5
-1y + 6 = -5
-1y = -11
y = 11
Step Three: To find the value of x, substitute 11 for y in the revised Equation
2 and simplify. x = -3y +3; x = -30
Step Four: Check to see if (-30,11) is the solution by substituting -30 for x and
11 for y in each of the original equations.
A. Solve the following linear systems using substitution.
1.
y = -x + 2
-2x + y = 5
2. 4x + 3y = -2
x + 5y = -9
3. 3x + y = -2
5x + 2y = -2
4. x = -2y + 2
x – 4y = 14
B. Real-Life Application Problems (Select One.)
Read the problem carefully. Highlight important information that may help
you understand the problem. Define your variables. Ask yourself “What am
I trying to solve for?” Write a verbal model for the situation, and then
translate that verbal model into an algebraic model. Remember that when
there are two variables, typically, you will need to write a system of
equations. Solve your algebraic model. Lastly, make sure that you answer
the question that is being asked.
1. The Smith family made an $800 down payment and pays $75 a month for new
furniture. At the same time, the Cooper family made a $500 down payment and
pays $95 a month for the same furniture. Use algebra to determine how many
months it will be before the amounts they have paid are equal.
2. In one day, the National Civil Rights Museum in Memphis, Tennessee, collected
$1590 from 321 people admitted to the museum. The price for each adult
admission was $6. People who were between the ages of 4 and 17 paid the child
admission price of $4. How many adults and how many children were admitted
on that day?
3. You are selling tickets for a high school play. Student tickets cost $4 and
general admission tickets cost $6. You sell 525 tickets and collect $2876. How
many of each type of ticket did you sell?
4. You are ordering softballs for two softball leagues online. The Pony League uses
an 11-inch softball priced at $2.75. The Junior League uses a 12-inch softball
priced at $3.25. You printed the receipt for the order; but unfortunately you
dropped it and it smeared in the rain. You recall, however, that you ordered 80
softballs for a total of $245. How many of each size did you order?
5. Your math teacher tells you that next week’s test is worth 100 points and
contains 38 problems. Each problem is worth either 5 points or 2 points.
Because you are studying systems of linear equations, your teacher says that
for extra credit you can figure out how many problems of each value are on the
test. How many problems of each value are there?
6. The value of your EFG stock is three times the value of your PQR stock. If the
total value of both stocks is $4500, how much is invested in each company?
Tier 2 Worksheet - Guided Practice
A. Solve the following linear systems using substitution.
1. 4x + 3y = -2
x + 5y = -9
2. 3x + y = -2
5x + 2y = -2
3.
x - 7y = 6
-3x + 21y = -18
B. Real-Life Application Problems (Select Three.)
1.
The Smith family made an $800 down payment and pays $75 a month for new furniture.
At the same time, the Cooper family made a $500 down payment and pays $95 a month
for the same furniture. Use algebra to determine how many months it will be before
the amounts they have paid are equal.
2. In one day, the National Civil Rights Museum in Memphis, Tennessee, collected $1590
from 321 people admitted to the museum. The price for each adult admission was $6.
People who were between the ages of 4 and 17 paid the child admission price of $4.
How many adults and how many children were admitted on that day?
3. You are selling tickets for a high school play. Student tickets cost $4 and general
admission tickets cost $6. You sell 525 tickets and collect $2876. How many of each
type of ticket did you sell?
4. You are ordering softballs for two softball leagues online. The Pony League uses an 11inch softball priced at $2.75. The Junior League uses a 12-inch softball priced at
$3.25. You printed the receipt for the order; but unfortunately you dropped it and it
smeared in the rain. You recall, however, that you ordered 80 softballs for a total of
$245. How many of each size did you order?
5. Your math teacher tells you that next week’s test is worth 100 points and contains 38
problems. Each problem is worth either 5 points or 2 points. Because you are studying
systems of linear equations, your teacher says that for extra credit you can figure out
how many problems of each value are on the test. How many problems of each value are
there?
6.
The value of your EFG stock is three times the value of your PQR stock. If the total
value of both stocks is $4500, how much is invested in each company?
Tier 3 Worksheet - Guided Practice
A. Solve the following linear systems using substitution.
1. 4x + 3y = -2
x + 5y = -9
2.
3.
5
x - y = -4
2
1
5x - 2y =
4
x-
2
y = -5
3
3x - 2y = -15
B. Real-Life Application Problems (Select 3 problems from #1–6, plus do #7.)
1. The Smith family made an $800 down payment and pays $75 a month for new
furniture. At the same time, the Cooper family made a $500 down payment and
pays $95 a month for the same furniture. Use algebra to determine how many
months it will be before the amounts they have paid are equal.
2. In one day, the National Civil Rights Museum in Memphis, Tennessee, collected
$1590 from 321 people admitted to the museum. The price for each adult
admission was $6. People who were between the ages of 4 and 17 paid the child
admission price of $4. How many adults and how many children were admitted
on that day?
3. You are selling tickets for a high school play. Student tickets cost $4 and
general admission tickets cost $6. You sell 525 tickets and collect $2876. How
many of each type of ticket did you sell?
4. You are ordering softballs for two softball leagues online. The Pony League uses
an 11-inch softball priced at $2.75. The Junior League uses a 12-inch softball
priced at $3.25. You printed the receipt for the order; but unfortunately you
dropped it and it smeared in the rain. You recall, however, that you ordered 80
softballs for a total of $245. How many of each size did you order?
5. Your math teacher tells you that next week’s test is worth 100 points and
contains 38 problems. Each problem is worth either 5 points or 2 points.
Because you are studying systems of linear equations, your teacher says that
for extra credit you can figure out how many problems of each value are on the
test. How many problems of each value are there?
6. The value of your EFG stock is three times the value of your PQR stock. If the
total value of both stocks is $4500, how much is invested in each company?
7. The price of refrigerator A is $600, and the price of refrigerator B is $1200.
The cost of electricity needed to operate the refrigerator is $50 per year for
refrigerator A and $40 per year for refrigerator B.
a) Writing Equations. Write an equation for the cost of owning
refrigerator A and an equation for the cost of owning refrigerator B.
b) Graphing Equations. Graph the equations from part (a).
c) Using Substitution. Solve the system of equations using substitution.
d) After how many years are the total costs of owning the refrigerators
equal?
e) Checking Reasonableness. Is your solution from graphing consistent with
your solution from substitution?
f) Is your solution reasonable in this situation? Explain.
Math Homework: Tic-Tac-Toe Choices (pp. xxx-xxx)
Everyone must go through the center.
Solve any 3 problems:
1-9
Solve any 3 problems:
1-9
Solve any 3 problems:
10-18
Solve any 3 problems:
10-18
Solve any 4 problems:
28-35
Solve any 3 problems:
1-9
Solve any 3 problems:
1-9
Solve any 3 problems:
19 - 27
Solve any 3 problems:
19-27
Tic –Tac –Toe Homework Practice Exercises
Directions: Everyone has to go through the middle. Show work on separate paper.
Check whether the given ordered
pairs are solutions:
a) 2x - 3y ≤ 2
(0,-1), (3,2)
b) 3x – 10y < -8 (6,3), (-4,-2)
Graph the inequality in a coordinate
plane:
a)
b)
-3x > 9
y ≤ -3
Graph the inequality in a
coordinate plane:
a)
b)
x > -3
-5 y ≥ 10
1. Graph the inequality in a
coordinate plane:
Check whether the given ordered
pairs are solutions:
a) 5x + y ≥ -3
(-3,6), (2,-5)
b) 4y – 2x < 5
(2,0), (-3,1)
Check whether the given ordered
pairs are solutions:
a) 6 x – 3y > -3
a) 2y + x ≥ 3
(-1,-2), (1,1)
b)
b) x < 2y + 5
(4,0), (-4,-5)
12x + 4y ≥ 8
2. The cheerleading group at
Concord is planning a
fundraiser. The group wants
to sell caps and t-shirts with
the school’s logo on them and
sell them at a profit. They
can sell caps for $10 each and
t-shirts for $15 each.
a) Write and graph an
inequality to describe how
many caps and t-shirts they
must sell to exceed $1800.
b) Explain how you can modify
this inequality to describe how
many caps and t-shirts they
must sell to exceed $600 in
profit if they made a 30%
profit on caps and 40% profit
on t-shirts.
c) Suppose they sold 50 caps
and 90 t-shirts. What point
on the coordinate plane would
represent this situation?
Graph the inequality in a coordinate
plane:
a)
-2y < 4
b)
x≥-
1
2
Check whether the given
ordered pairs are solutions:
a) 2y + x ≥ 3 (-1,-2), (1,1)
b) x + 2y > 4
(2,1), (-3,6)
Graph the inequality in a
coordinate plane:
a)
b)
2x > 8
y ≥ -2
“Off to the Races” Cooperative Learning Review
1. Form tiered teams of four students per team. If there are leftover students,
allow them to monitor/assist in the execution of the cooperative review
process.
2. Each team is seated in a row. Students are placed in seats according to their
math readiness, with the first person in the row having the lowest capability in
the subject matter and the fourth person having the highest capability.
3. Each person in the first three seats completes a portion of a group problem,
and then transfers his/her response to the person seated behind them. The
goal is to funnel interim information down the line so that the last person
(Person #4) is in a position to put all the pieces together to answer the
capstone question correctly. When Person #4 gets an answer, he/she “races”
to the board to write that answer on the board. The monitor says only “yes” or
“no”. If the monitor says “no”, everyone keeps working on the problem. A new
answer can be given if the mistake is found.
4. Each student has a life-line, i.e., they are allowed to ask the person in front or
behind them one question at a time: “Is this correct?” The answer is either
“yes” or “no”. The person cannot explain or help in any other fashion. No one
may leave his/her seat to assist another student.
5. Whichever team answers the final capstone question the quickest is the winner.
Each person demonstrates how they solved his/her piece of the race. Each
person on the winning team receives one extra credit/bonus point on the
upcoming test/quiz.
“Off to the Races” … Cooperative Review Example for Algebra 1 Test
Race One
1.
Solve the equation: 16 = x – (–9)
2. Solve the equation: 40y = –160
3. Solve the equation:
4. Simplify/Evaluate:
3
p = 3
8
21
*y–p
x
--------------------------------Race Two
1. Solve the equation: 6a – 7 – 2a = – 35
2. Solve the equation: 3(2b – 2) + 3b = 2 (5b – 8)
3. Solve the equation:
0.15x – 4.5 = 0.3
4. Simplify/Evaluate:
2a + b * x
----------------------------------Race Three
1. Solve the equation:
2
1
5
x–
=
7
2
14
2. Rewrite so that y is a function of x (i.e, solve for y):
1
(y + 5) + 4x = 3x
2
3. Solve the equation:
0.03c – 1.2c = – 4.68
4. Simplify/Evaluate:
3x + y –
1
c
2
Cooperative Learning Jigsaw Activity
Just as in the “Off to the Races” Cooperative Review, each student’s part to this jigsaw
activity is essential for the completion and full understanding of the final project.
Students work together as a team toward the target material, especially when that
material contains several chunks of related information.
Step 1 – Identify your target material, e.g., understand transformations of trig functions
if given 𝑓(𝑥) = a sin(𝑏𝑥 − 𝑐) + 𝑑
Step 2 – Determine the number of pieces (i.e., chunks of information) needed to complete
the puzzle.
Step 3 – Divide the class into groups (jigsaws) of three or four depending on the size of
the class and the number of pieces for the puzzle. Give each group a home number (e.g.,
Group 1, Group 2, etc.) Have each group sit together and elect a group leader. Let them
know that each of them will become an expert on some aspect of the question so that they
can “teach” the other members of their jigsaw group.
Step 4 – To form the expert groups, you can do it randomly or use your own grouping
strategy based on math readiness. Ask students to relocate to those expert groups, but
make sure that they remember their original home group number.
Step 5 – Visit each expert group and give each person a handout containing the numbered
or lettered piece of the puzzle. Each expert group is expected to develop an
understanding and address any questions on their particular topic.
For example, in the trig transformation activity: Expert A develops an understanding of
amplitude of the motion, Expert B develops an understanding of rate of motion and how it
affects the period of the graph, Expert C develops an understanding of horizontal
shifting, and Expert D develops an understanding of vertical shifting.
Remind students that they should take notes and address any questions on the handouts so
that they can go back to their home groups and teach them what they have learned.
Step 6 – After an appropriate amount of time, ask students to reassemble in their home
groups. Each group leader then asks each expert to share insights on their individual
piece. The goal is to assemble the jigsaw puzzle completely and for each person on the
team to see the overall picture.
Step 7 – Do an informal assessment by asking each group to summarize their learnings
about the subject matter or solve a problem which involves all pieces of the puzzle.
Project–Based / Self-Discovery Project
Regression Analysis (Optional)
Many situations in real life involve finding relationships between two variables.
Generally, this requires the collection and analysis of data. A regression model
can typically be used to represent the relationship between those variables. For
example, there is generally a relationship between the number of hours that
students spend studying for a test and test grades. Think about some of these
situations and collect data on one of them. You may find that the internet,
newspapers, and magazines may be useful as you brainstorm ideas or gather data.
You will need to collect at least 20 data points and make a scatter point on graph
paper. You will also need to find an equation of the best fit and the correlation
coefficient. You may use your calculator to find both. In addition, you must
answer the following:
Is the correlation positive or negative?
How strong is the correlation?
Is your equation a good predictor?
Where did you collect your data?
What were your learnings?
A completed project should also include the data table, the scatter plot, the
regression equation, the correlation coefficient, the answered questions, and a
class presentation of your project. Lastly, all students must use a large poster
board to display project findings. The due date is Thursday, October X, during our
scheduled class time. No exceptions. The project is worth 10 extra credit points.
Students will be assessed according to a completed project rubric.