Unit 5
Proportional and Non-Proportional Functions
8th Grade
5E Lesson Plan Math
Grade Level: 8
Lesson Title: Unit 05 - Proportional
and Non-Proportional Functions
THE TEACHING PROCESS
Subject Area: Math
Lesson Length: 17 days
Lesson Overview This unit bundles student expectations that address problems
involving proportional and non-proportional situations, direct variation, identifying
functions, saving for college, and the effect of long-term investments. According to
the Texas Education Agency, mathematical process standards including
application, a problem-solving model, tools and techniques, communication,
representations, relationships, and justifications should be integrated (when
applicable) with content knowledge and skills so that students are prepared to use
mathematics in everyday life, society, and the workplace.
During this unit, students extend their previous understandings of slope and yintercept to represent proportional and non-proportional linear situations with
tables, graphs, and equations. These representations are used as students
distinguish between proportional and non-proportional linear situations. Students
specifically examine the relationship between the unit rate and slope of a line that
represents a proportional linear situation. Problem situations involving direct
variation are included within this unit as they are also proportional linear situations.
Graphical representations of linear equations are examined closely as students
begin to develop the understandings of systems of equations. Students are
expected to identify the values of x and y that simultaneously satisfy two linear
equations in the form y = mx + b from the intersections of the graphed equations.
Students must also verify these values algebraically with the equations that
represent the two graphed linear equations. The study of proportional and nonproportional linear situations allows students to enrich their understanding of
financial situations by explaining how small amounts of money, without interest,
invested regularly grow over time. Students also examine how periodic savings
plans can be used to contribute to the cost of attending a two-year or four-year
college after estimating the financial costs associated with obtaining a college
education. Students are formally introduced to functions as a relation in which
each element of the input (x) is paired with exactly one element of the output (y).
Students must identify functions using sets of ordered pairs, tables, mappings, and
graphs. Examining proportional and non-proportional linear relationships is
extended to include identifying proportional and non-proportional linear functions in
mathematical and real-world problems. A deep understanding of the
characteristics of functions is essential to future mathematics coursework beyond
Grade 8.
Unit Objectives:
Students will:
 extend their previous understandings of slope and y-intercept to represent
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Unit 5
Proportional and Non-Proportional Functions
8th Grade
proportional and non-proportional linear situations with tables, graphs, and
equations
 distinguish between proportional and non-proportional linear situations
 examine the relationship between the unit rate and slope of a line that
represents a proportional linear situation
 develop the understandings of systems of equations
 recognize problem situations involving direct variation & graphical
representations of linear equations as proportional or non-proportional
relationships
 identify the values of x and y that simultaneously satisfy two linear
equations in the form y = mx + b from the intersections of the graphed
equations.
 verify these values algebraically with the equations that represent the two
graphed linear equations
 enrich their understanding of financial situations by explaining how small
amounts of money, without interest, invested regularly grow over time
 examine how periodic savings plans can be used to contribute to the cost of
attending a two-year or four-year college after estimating the financial costs
associated with obtaining a college education
 identify functions using sets of ordered pairs, tables, mappings, and graphs
 identifying proportional and non-proportional linear functions and
relationships in mathematical and real-world problems
Standards addressed:
TEKS: 8.1A - Apply mathematics to problems arising in everyday life, society, and
the workplace.
8.1B – Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution.
8.1C - Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation, and
number sense as appropriate, to solve problems.
8.1D - Communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams, graphs, and language as
appropriate.
8.1E - Create and use representations to organize, record, and communicate
mathematical ideas.
8.1F - Analyze mathematical relationships to connect and communicate
mathematical ideas.
8.1G - Display, explain, and justify mathematical ideas and arguments using
precise mathematical language in written or oral communication.
8.4B – (READINESS STANDARD) - Graph proportional relationships, interpreting
the unit rate as the slope of the line that models the relationship.
8.5A – (SUPPORTING STANDARD) - Represent linear proportional situations with
tables, graphs, and equations in the form of y = kx.
8.5B - (SUPPORTING STANDARD) - Represent linear non-proportional situations
with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
8.5E - (SUPPORTING STANDARD) - Solve problems involving direct variation.
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Proportional and Non-Proportional Functions
8th Grade
8.5F - (SUPPORTING STANDARD) - Distinguish between proportional and nonproportional situations using tables, graphs, and equations in the form y = kx or y =
mx + b, where b ≠ 0.
8.5G - (READINESS STANDARD) - Identify functions using sets of ordered pairs,
tables, mappings, and graphs.
8.5H - (SUPPORTING STANDARD) - Identify examples of proportional and nonproportional functions that arise from mathematical and real-world problems.
8.9A - (SUPPORTING STANDARD) - Identify and verify the values of x and y that
simultaneously satisfy two linear equations in the form y = mx + b from the
intersections of the graphed equations.
8.12C - (SUPPORTING STANDARD) - Explain how small amounts of money
invested regularly, including money saved for college and retirement, grow over
time.
8.12G - (SUPPORTING STANDARD) - Estimate the cost of a two-year and fouryear college education, including family contribution, and devise a periodic savings
plan for accumulating the money needed to contribute to the total cost of
attendance for at least the first year of college.
ELPS:
Misconceptions:
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Some students may not relate the constant rate of change or unit rate
to m in the equation y = mx + b.
Some students may not relate the constant of proportionality or unit rate
as k in the equation y = kx or m in the equation y = mx + b, when b = 0.
Some students may think that a constant rate of change always means the
situation is always proportional.
Some students may not associate slope represented as whole number as a
rational number that can be represented as .
Some students may think that a function can have multiple outputs (y) for
the same input (x).
Some students may think that a function cannot have multiple inputs (x) that
correspond to the same output (y).
Some students may think that the slope in a linear relationship
is
, since the x-coordinate (horizontal) always comes before
the y-coordinate (vertical) in an ordered pair, instead of the correct
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representation that slope in a linear relationship is
.
Some students may think that the intercept coordinate is the zero term
instead of the non-zero term, since intercepts are associated with zeros. In
other words, students may think (0, 4) would be the x-intercept because the
0 is in the x coordinate.
Students may not graph lines correctly on the coordinate plane
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Unit 5
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Proportional and Non-Proportional Functions
Students may use (y,x) as the ordered pair instead of (x,y)
8th Grade
Vocabulary:
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401(k) – a set amount of money, or percentage of pay, that is set aside
from an employee’s pay check by their employer, before the employee’s
wages are taxed. The employer may or may not contribute as well to the
employee’s 401(k) fund depending on employer’s policy. The money is
taxed when it is withdrawn at retirement age. In addition, if withdrawn prior
to retirement age, an additional penalty tax is assessed.
403(b) – a set amount of money, or percentage of pay, that is set aside
from an employee’s pay check by their employer, before the employee’s
wages are taxed. The money is taxed when it is withdrawn at retirement
age. In addition, if withdrawn prior to retirement age, an additional penalty
tax is assessed.
529 account – educational savings account managed by the state, and is
usually tax-deferred
Annuity – deductible and non-deductible contributions may be made, taxes
may be waived if used for higher education
Direct subsidized federal student loan – a loan issued by the U.S.
government in an amount determined by the college available to
undergraduate students who demonstrate a financial need where the U.S.
Government pays the interest on the loans while the student is enrolled at
least half-time, up to six months after leaving school, or during a requested
deferment period
Direct unsubsidized federal student loan – a loan issued by the U.S.
government in an amount determined by the college available to
undergraduate or graduate students where the interest is paid by the
borrower from the time the loan is initiated, even during requested
deferment or forbearance periods
Direct variation – a linear relationship between two
variables, x (independent) and y (dependent), that always has a constant
unchanged ratio, k, and can be represented by y = kx
Function – relation in which each element of the input (x) is paired with
exactly one element of the output (y)
Grant – money that is awarded to students usually based on need with no
obligation to repay this money
Individual retirement account (IRA) – a set amount of money, or
percentage of pay, that is invested by an individual with a bank, mutual
fund, or brokerage
Inflation – the general increase in prices and decrease in the purchasing
value of money
Linear relationship – a relationship with a constant rate of change
represented by a graph that forms a straight line
Principal – the original amount invested or borrowed
Private student loan – a loan issued by a lender other than the U.S.
Government
Retirement savings – optional savings plans or accounts to which the
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Proportional and Non-Proportional Functions
8th Grade
employer can make direct deposits of an amount deducted from the
employee's pay at the request of the employee
Savings account – a bank or credit union account in which the money
deposited earns interest so there will be more money in the future than
originally deposited
Scholarship – money that is awarded to students based on educational
achievement with no obligation to repay this money
Slope – rate of change in y (vertical) compared to the rate of change
in x (horizontal),
or
or
, denoted
as m in y = mx + b
Social Security – a percentage of an employee's pay required by law that
the employer withholds from the employee's pay for social security savings
which is deposited into the federal retirement system; payment toward that
employee's eventual retirement; the employer also is required to pay a
matching amount for the employee into the federal retirement system
Student loan – borrowed money that must be paid back with interest
Taxable investment account – many companies will create an investment
portfolio with the specific purpose of saving and building a strong portfolio to
be used to pay for college
Traditional savings accounts – money put into a savings account much
like paying a monthly expense such as a light bill or phone bill
U.S. savings bond – money saved for a specific length of time and
guaranteed by the federal government
Unit rate – a ratio between two different units where one of the terms is 1
Work study – programs that allow students to work in exchange for a
portion of their tuition
y-intercept – y-coordinate of a point at which the relationship crosses
the y-axis meaning the x-coordinate is equal to zero, denoted
as b in y = mx + b
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Unit 5
Proportional and Non-Proportional Functions
8th Grade
List of Materials: Activity Sheets (all links attached at the end of the lesson),
calculator
INSTRUCTIONAL SEQUENCE
Phase: Engage
Day 1
Day 1 - Activity: Have students discuss the following questions in a group or with
a partner by displaying them on the board
Click on the link or scan the QR code to view the questions in “board display”
mode or to print the “Assessing prior knowledge” questions.
Intro to Functions
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Assess prior knowledge: What is a proportional relationship? How do we
identify a proportional relationship in an algebraic expression? On a
coordinate plane? In a table? On a graph? Is it always a straight line?
Assess prior knowledge: What is a non-proportional relationship? How do
we identify a non-proportional relationship in an algebraic expression? On
a coordinate plane? In a table? On a graph? Is it always a straight line?
Activity: Go over vocabulary. Link to print vocab: Day 1 Vocabulary
To prepare for tomorrow’s lesson, provide students with access to the internet (or
costs associated with attending a Texas college for one year) so they are able to
find the costs for attending a Texas college for one year.
What’s the teacher doing?
What are the students doing?
Teacher should be moving about the
room, listening to the explanations of
the above questions within the groups.
Note mathematical vocabulary and
understanding of concepts.
Students should be working in a group
or with a partner and explaining the
answers to the questions. Once one
student is done answering/finding the
answers to the questions, they should
switch roles and answer the questions
again until everyone has taken a turn.
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Questioning: What is a
proportional relationship? How
do we identify a proportional
relationship in an algebraic
expression? On a coordinate
plane? In a table? On a graph?
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Answering: A proportional
relationship is one that shows
direct variation, where y=kx, has
a constant of proportionality,
where k=y/x, or can be put into
fractions to equal the same
Unit 5
Proportional and Non-Proportional Functions
8th Grade
Is it always a straight line?
number. Proportional
relationships always show a
straight line, but a straight line
may not always show a
proportional relationship. On a
graph, it must start or go through
the origin and must be straight.
In an equation or expression,
there cannot be any adding or
subtracting, only multiplying or
dividing.
Phase: Explore
Day 2
Day 2 - Activity: Prior to computer lab use, print out the “College Expenses
Worksheet” for students or group to fill in (6 copies per page). College Expenses
Worksheet
Using a computer or mobile device, students will find the cost of attending a Texas
college for one year. The costs should include tuition, books, fees, boarding, and
living expenses.
Example: TAMU Costs
While in the lab, have students discover 529 accounts and create a short
presentation of an explanation in their own words.
Questioning:
 What is tax-deferred?
 How does a 529 account compare to a traditional savings account?
 How much can be contributed into a 529 account?
 Why is education planning important?
Present findings to the class.
For the teacher: Example Explanation of 529 Plan/Account
Day 3
More Exploring: Make a foldable of the different parts of y=mx+b Ex: Slope
Intercept Foldable
Activity: Teams of students are assigned a task that asks them to gather and
record data sets of related quantities. They devise a plan and then collect and
organize data. They use the data sets to determine whether or not there is a
functional relationship between the quantities and explain why. Finally, each team
presents their task, their plan, and the data they collected to the class.
Example tasks:
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Students gather data about the lengths of the first name and last name for
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Unit 5
Proportional and Non-Proportional Functions
8th Grade
each member of their class. They record the information in a table. They
then determine that the relationship between the length of a student’s first
name and last name is not a functional relationship. Two students with a first
name of equal length may have last names of a different length, so the
length of a last name cannot be predicted using the length of the first name.
 Students gather data on exam scores and the number of hours students
studied for the exam. While these should correlate (the more time a student
studies the more likely his/her score is high), the relationship may not be
functional. Data may reveal that two students studying the same amount of
time earned different scores.
 Students gather data relating the height of a stack of identical cups to the
number of cups in the stack. They record the information in a table. They
then determine that the relationship between the height of a stack of cups
and the number of cups in the stack is a functional relationship; that is, stack
height depends on the number of cups in the stack.
Assessment Connections
Ask . . .
Start with . . .
 Tell me about the data you collected.
Probe further with . . .
 How did you organize the data? (in a graph, table, equation). Why did you
choose to organize it in this way? (looks neater, to see the relationships)
What units of measure did you use? Do the units of measure make sense?
 What is a function? (something with x & y, one input for every output)
 Does one quantity depend on the other in a systematic way? How do you
know? (because the input changes the output)
 Describe the relationship between the variables using a function statement.
(as x increases, so does y or vice-versa)
 What is the independent/dependent quantity? How do you know?
Day 4
Activity: Students are given a problem situation that can be described using a
functional relationship. They determine that the relationship is a function and
represent the function using a variety of ways, including a function rule. They
answer questions arising from the situation by writing and solving equations or
inequalities.
Two example situations . . .
The functional relationship:
Scott lives in Texas. His friend lives in Michigan. Scott has to make a long
distance call to talk to his friend. He plans to use his calling card. The card
charges $0.75 for access to long distance plus $0.10 for each minute.
Determine the functional relationship that describes how the total charge
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depends on the length of a call in minutes.
8th Grade
Answering questions by writing and solving equations or inequalities:
Scott lives in Texas. His friend lives in Michigan. Scott has to make a long
distance call to talk to his friend. He plans to use his calling card. The card
charges $0.75 for access to long distance plus $0.10 for each minute. He has
$3.00 left on his card. Investigate whether or not Scott can talk with his friend
for at least 15 minutes.
What’s the teacher doing?
Teacher should be reminding students
of proper vocabulary use while creating
presentation. Teacher should be
searching for specifics in the
student/group presentation that shows
relationship to prior knowledge.
Teacher is questioning students
throughout the process, talking about
how the information may be presented
in a graph and table, as multiple forms
of representation.
What are the student’s doing?
Working in groups to create
presentation over cost of attending a
Texas college for one year.
Listen for . . .
 Can the student determine
whether or not a relationship is a
functional relationship?
 Does the student clearly and
accurately describe what it means
for a relationship between two
quantities to be a functional
relationship?
 Does the student use appropriate
language when describing the
functional relationship?
 Does the student appropriately
identify independent and
dependent quantities for the
functional relationship described?
Look for . . .
 Does the student demonstrate that
he/she understands that a function
represents a dependence of one
quantity on another?
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Unit 5
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Proportional and Non-Proportional Functions
8th Grade
Can the student use data sets to
determine whether or not a
relationship is a function?
Can the student organize and
record the collected data in an
efficient and useful way (for
example, length of name and
month of birth of same person)?
Does the student accurately
represent the data? Check for
reasonable units of measure.
Does the student label the data
correctly?
Phase: Explain
Day 5 - Activity: To further explain equations, do the following activities.
Throughout the unit, practice defining variables and translating word problems into
equations, then creating a table of values and graphing the equation to show the
relationship among the representations. Do this by using Rules 4 Equations
Worksheet.
Worksheet: Rule 4 Equations
Day 6 - Activity: Print each of the 20 problems on an 8.5 x 11 piece of paper,
spread out all over the library, classroom, or hallway. The students get the answer
sheet and rotate through the problems in any order to write and interpret the linear
equations. The link includes the answer sheet and the 2 pages that have all 20
problems on them (2 days).
Worksheet: Desk Hop
Day 7 – Activity 1: Teacher should have students create graphs using the
following cards with equations. One set per group of 4 students. Present their
graphs to the class once completed. Equation Cards for Graph
Activity 2: Print the worksheet by clicking on the link. Work the first 2 problems
during a class discussion. This activity focuses on the intercept of 2 functions
when graphing. Students will be able to identify and verify the values of x and y
that simultaneously satisfy two linear equations in the form of y=mx+b.
Intercepting Lines on a Graph
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Unit 5
Proportional and Non-Proportional Functions
What’s the teacher doing?
What are the students doing?
Teacher is evaluating the progress of
students and assessing their
performance. Teacher is listening to use
of vocabulary throughout the lesson
while encouraging and monitoring
students to stay on task and work with
group.
8th Grade
Working with group, using vocabulary
list to discuss and explain to other
group members. Students should be
prepared to present to the class to show
their understanding.
Phase: Elaborate
Day 8 - Activity: Teacher will have students fill in the following y = mx+b sheet.
One per student. Fill in the Equation
What’s the teacher doing?
What are the students doing?
Teacher is evaluating the progress of
students and assessing their
performance. Teacher is listening to use
of vocabulary throughout the lesson
while encouraging and monitoring
students to stay on task and work with
group.
Provided worksheet.
Phase: Evaluate
Day 8 Continued - Activity: Performance Assessment #1 Board Display of
Activity
Analyze the problem situation(s) described below. Organize and record your work
for each of the following tasks. Using precise mathematical language, justify and
explain each solution process.
1) Bennett noticed a beetle crawling towards him. The table below shows the
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distance the beetle was from him after each second.
8th Grade
a) Use the data in the table to graph the relationship between the number of
seconds and distance traveled.
b) Use the table of data or graph to determine the rate of change, or slope, and yintercept and explain what each of them represents in the context of the problem
situation.
c) Write an equation to represent the problem situation where x represents the
time in seconds and y represents the distance the beetle is from Bennett.
d) Use the graph to describe how similar right triangles can be used to justify how
the slope of the line representing the problem situation is the same for any two
points on the line.
What’s the teacher doing?
What are the students doing?
Teacher is evaluating the progress of
students and assessing their
performance. Teacher is listening to use
of vocabulary throughout the lesson
while encouraging and monitoring
students to stay on task and work with
group.
Provided Performance Assessment
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Phase: Engage – Unit 5 Lesson 2
Interest
Day 9 - Activity: Display the Student Loan table below and explain to
students the different parts of the table, such as the interest, time, and
principal. Display Student Loan Table
Have students use the TVM solver on the TI-84. Directions: TVM Solver
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8th Grade
Unit 5
Proportional and Non-Proportional Functions
Phase: Explore/Explain/Elaborate
Days 9-15
8th Grade
The attached print out covers the 3 phases of Exploring, Explaining, and
Elaborating.
Compare investments and use the card sets to match to the different investments.
Match the “Tables” to the “Graphs” and create “Formulas” using the
interest/investment cards.
Use the “Making Money” sheet (included in the attachment) and work
collaboratively with other group members to answer the questions over graphs,
tables, equations, interest, and investments.
Investments Lesson
This lesson shows simple savings and interest.
Phase: Evaluate
Days 16 & 17 Activity 1 & 2
Performance Assessment #2
Activity 1
Analyze the problem situation(s) described below. Organize and record your work
for each of the following tasks. Using precise mathematical language, justify and
explain each solution process.
1) From the day she was born, Elle’s parents began saving money for her to
attend a Texas college when she graduated from high school. In preparation for
that event, her parents deposited $750 a year since she was born into a special
savings account for her education.
a) Without the consideration of interest earned on the account, generate a table,
graph, and equation that can be used to represent this situation in terms of x,
number of years, and y, the balance of the savings account.
b) Using your generated graph, describe how the unit rate of the amount of money
deposited annually is related to the slope of the line.
c) Describe if the relationship represented in the table, graph, and equation
represents a proportional or non-proportional situation.
d) Determine how much money will be contained in the account once the annual
amount is deposited in the savings account when Elle turns 18.
e) Describe how the number of years and amount of money Elle’s parents
regularly deposit affects the total amount of money invested for her college
education.
2) Using the Internet or another resource, research the costs associated with
attending a Texas two-year versus four-year college.
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a) Determine if Elle will have enough money saved for the first year of college at
either type of college if her parents deposit $750 in a savings account each year
for 18 years.
b) Devise a periodic savings plan that Elle and her family can use to ensure she
will have enough money to attend the first year of attendance at a Texas four-year
college.
Performance Assessment #3
Activity 2
For this activity, students will need to create their graphical representations
on the same coordinate plane.
Analyze the problem situation(s) described below. Organize and record your work
for each of the following tasks. Using precise mathematical language, justify and
explain each solution process.
1) From the day she was born, Elle’s parents began saving for her to attend a
Texas college when she graduated from high school. In preparation for that event,
her parents deposited $750 a year since she was born into a special savings
account for her education. Her grandparents also deposited $27,250 into the
savings account on the day that Elle was born.
a) Without the consideration of interest earned on the account, generate a table,
graph, and equation that can be used to represent this situation in terms of x,
number of years, and y, the balance of the savings account.
b) Describe if the relationship represented in the table, graph, and equation
represents a proportional or non-proportional situation.
c) Use a set of ordered pairs, table, mapping, or graph to determine if the
relationship represents a function.
2) From the day he was born, Brandon’s parents began saving for him to attend a
Texas college when he graduated from high school. In preparation for that event
his parents deposited $2,250 a year since he was born into a special savings
account for his education. His grandparents deposited $1,750 into the savings
account on the day that Brandon was born
a) Without the consideration of interest earned on the account, generate a table,
graph, and equation that can be used to represent this situation in terms of x,
number of years, and y, the balance of the savings account.
b) Describe if the relationship represented in the table, graph, and equation
represents a proportional or non-proportional function.
3) At one point, the age of Elle and Brandon as well as the total amount of money
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deposited in the savings account will be equivalent.
a) Use the graphs of your equations to represent Elle and Brandon’s savings
account balances to determine the point of intersection that simultaneously
satisfies both linear equations.
b) Describe the meaning of the point of intersection in the context of the situation.
c) Describe the relationship between the number of years money is deposited into
the savings account and the amount of money deposited into the savings account.
What’s the teacher doing?
What are the students doing?
Monitoring students while starting a new Using accurate mathematical language
lesson regarding investments. Listening regarding interest, banking,
to proper mathematical language.
scholarships, etc.
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Daily Materials
Day 1
Intro to Functions
Day 1 Vocabulary
Day 2
TAMU Costs
College Expenses Worksheet
Example Explanation of 529 Plan/Account
Day 3
Slope Intercept Foldable
Day 5
Rule 4 Equations
Day 6
Desk Hop
Day 7
Equation Cards for Graph
Intercepting Lines on a Graph
Day 8
Fill in the Equation
Board Display of Activity
Days 9-15
Investments Lesson
Days 16 & 17
Display Student Loan Table
TVM Solver
Days 16 & 17 Activity 1 & 2
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8th Grade