Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
5E Lesson Plan Math
Grade Level: 8th
Lesson Title: One-Variable
Equations, Inequalities, and their
Applications
THE TEACHING PROCESS
Subject Area: Mathematics
Lesson Length: 13 Days
Lesson Overview
This unit bundles student expectations that address modeling, writing, and solving
one variable equations and inequalities with variables on both sides of the equal
sign or inequality sign using rational number coefficients and constants; and
solving problems comparing interest rates on loans and savings accounts, loan
lengths and total cost of repaying a loan. 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 understanding of solving equations to model
and solve one-variable equations with variables on both sides of the equal sign.
Students use their knowledge of equations to write one-variable equations or
inequalities with variables on both sides to represent problems using rational
numbers coefficients and constants. When given a one-variable equation or
inequality with variables on both sides, students are expected to write a
corresponding real-world problem. Financial literacy contexts such as calculating
and comparing simple and compound interest rates and how those rates affect
earnings in a savings account or the total cost of repaying a loan or credit card are
embedded in this unit.
Unit Objectives:
Students will…
extend their understanding of solving equations to model and solve one-variable
equations with variables on both sides of the equal sign,
use their knowledge of equations to write one-variable equations or inequalities
with variables on both sides to represent problems using rational numbers
coefficients and constants,
write a corresponding real-world problem when given a one-variable equation or
inequality with variables on both sides,
calculate/compare simple and compound interest rates and how those rates
affect earnings in a savings account or the total cost of repaying a loan or credit
card,
solve real-world problems comparing how interest rate and loan length affect the
cost of credit.
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Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
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.
8.8A Write one-variable equations or inequalities with variables on both sides that
represent problems using rational number coefficients and constants.
8.8B Write a corresponding real-world problem when given a one-variable
equation or inequality with variables on both sides of the equal sign using rational
number coefficients and constants.
8.8C Model and solve one-variable equations with variables on both sides of the
equal sign that represent mathematical and real-world problems using rational
number coefficients and constants.
8.12A Solve real-world problems comparing how interest rate and loan length
affect the cost of credit.
8.12B Calculate the total cost of repaying a loan, including credit cards and easy
access loans, under various rates of interest and over different periods using an
online calculator.
8.12D Calculate and compare simple interest and compound interest earnings.
ELPS:
ELPS.c.1A use prior knowledge and experiences to understand meanings in
English
ELPS.c.2D monitor understanding of spoken language during classroom
instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm
understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio
tape, video, DVD, and CD ROM to build and reinforce concept and language
attainment
ELPS.c.2G understand the general meaning, main points, and important details of
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Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
spoken language ranging from situations in which topics, language, and contexts
are familiar to unfamiliar
ELPS.c.3D speak using grade-level content area vocabulary in context to
internalize new English words and build academic language proficiency
ELPS.c.3J respond orally to information presented in a wide variety of print,
electronic, audio, and visual media to build and reinforce concept and language
attainment.
ELPS.c.4F use visual and contextual support and support from peers and
teachers to read grade-appropriate content area text, enhance and confirm
understanding, and develop vocabulary, grasp of language structures, and
background knowledge needed to comprehend increasingly challenging language
ELPS.c.4H read silently with increasing ease and comprehension for longer
periods
ELPS.c.4J demonstrate English comprehension and expand reading skills by
employing inferential skills such as predicting, making connections between ideas,
drawing inferences and conclusions from text and graphic sources, and finding
supporting text evidence commensurate with content area needs
ELPS.c.5B write using newly acquired basic vocabulary and content-based gradelevel vocabulary
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns,
and connecting words to combine phrases, clauses, and sentences in increasingly
accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to
fulfill content area writing needs as more English is acquired.
Misconceptions:
•
Some students may think that anytime a negative is involved, the
inequality switches rather than applying that rule when dividing or
multiplying both sides by a negative.
•
Some students may think that a constant term can be combined with
a variable term (e.g., 2x + 5 = 7x).
Underdeveloped Concepts:
•
Some students may think that answers to both equations and
inequalities are exact answers instead of correctly identifying the solutions
to equations as exact answers and the solutions to inequalities as range of
answers.
•
Some students may think variables are letters representing an object
as opposed to representing a number or quantity of objects.
•
Some students may think the equal sign means, “solve this” or “the
answer is” rather than understanding that equal sign represents a
quantitative and balanced relationship.
Vocabulary:
pg. 3
Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
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Amortization – process of paying down a loan with payments
that include both principal and interest until the full amount of the loan
is paid in full
•
Annual percentage rate (APR) – annual percentage rate applied to
the balance on a loan compounded monthly
•
Car title loan – a high-interest, short term loan of cash for which an
automobile title is required as collateral
•
Coefficient – a number that is multiplied by a variable(s)
•
Collateral – something which is pledged to secure repayment of a
loan; in the event of default on the loan, the collateral is forfeited
•
Compound interest – interest that is computed on the latest balance,
including any previously earned interest that has been added to the original
principal
•
Constant – a fixed value that does not appear with a variable(s)
•
Consumer loan – loans made by various businesses and financial
institutions
•
Credit – buying or obtaining goods or services now with an
agreement to pay in the future
•
Equation – a mathematical statement composed of algebraic and/or
numeric expressions set equal to each other
•
Inequality – a mathematical statement composed of algebraic and/or
numeric expressions set apart by an inequality symbol
•
Order of operations – the rules of which calculations are performed
first when simplifying an expression
•
Payday loan – a high-interest, short term loan of cash for which
collateral, such as an automobile title, is required
•
Principal – the original amount invested or borrowed
•
Simple interest – interest paid on the original principal in an account,
disregarding any previously earned interest
•
Solution set – a set of all values of the variable(s) that satisfy the
equation or inequality
•
Variable – a letter or symbol that represents a number
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pg. 4
Annual
Evaluate
Parentheses/brackets
Constraint
Forfeit
Rational number
Condition
Greater than
Repayment period
Balance
Greater than or equal to
Simplify
Default
Interest rate
Solve
Unit 3: One- Variable Equations, Inequalities, and their Applications
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8th Grade
Equality
Integer
Solution
Equal to
Less than
Title
Expression
Less than or equal to
Whole number
Exponent
Not equal to
List of Materials:
Bags/boxes – (used as variable representation)
Counters – (beans, chips, coins, marbles... anything really)
Individual student white boards, expo markers, erasers
Flash or index cards with inequality symbols
Day 1
Day 2-4
Day 5
Day 6-8
Day 9-11
Day 12
Day 13
counters
individual mini white boards for each student & expo markers
index cards
access to computers & internet
Grocery shopping & Vacation activities
continuation of Grocery & Vacation activities
access to computers & internet
assessment
Free worksheets for operations & inequalities can be obtained at
www.freemathworksheets.com
www.math-drills.com
www.mathworksheetsland.com
www.mathworksheets4kids.com
www.kutasoftware.com
Phase One: Engage the students
pg. 5
Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
Day 1 Activity: Students arrive in class and you hand them a handful of tokens,
chips, beans, or some other form of counters. On each of their desks is an empty
bag. Ask the students how many tokens they need in the bag to reach a certain
number (you decide on). (Various prizes can be used for encouragement)
Each student should have a different answer. Call on a select few students to
answer how many tokens they'll need in their bag to reach the number you chose.
As the students answer, you can correlate their answers with a formula on the
board, relating the bag as a variable (B for bag, X, etc... I would incorporate a
variety of variables to show them anything could be used).
20 = A + 5
A + 5 = 20
20 (number they need) = A (number in their bag) + 5 (number they were given)
Further this by breaking the students into pairs/groups and combining their bags.
Offer them a new number and explain that each bag must have the same number.
Examples:
3B + 14 = 32
3B = 18
each bag needs six
You can now release the students to work in groups or pairs answering
predetermined problems while you monitor.
After the students grasp the concept, select or take 2 volunteers and offer them
additional tokens and bags. Have the two compare what they have and ask them
how many tokens they'd need in their bags to be equal to one another. Remind
the students the bags need to have the same amount in each.
2X + 8 = x + 12
x=4
Work several more problems with other students before breaking them up in pairs
to work additional problems.
Introductions and explanations of new vocabulary and refreshing prior terminology
might be necessary. This can be done via a power point presentation or engaging
the students by randomly calling on them individually or as a group to answer.
Journals can also be used to copy and define definitions.
Reinforce order of operations and use model drawings to demonstrate guided
examples.
pg. 6
Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
Note: When reviewing prior material and concepts (note 2014-2015 will have a
gap in working with inequalities but should have previously solved for single
variable equations)
What’s the teacher doing?
What are the students doing?
Leading the activity, engaging the
students for participation and
encouragement.
Answering guided questions.
Participating in pairs/groups.
Phase Two: Explore the Concept
Days 2-4 (subtraction/multiplication/division – one operation per day?)
Activity: Mini White Boards - Used multiple times as each new operation is
introduced. Poster boards can be used as an alternative. (encourage students to
show their work, pairs and small groups can demonstrate problems through
equations and modeling)
Elaborate on the previous days lesson by introducing subtraction problems.
Engage the students by having them solve for the variables of problems you
provide on the board.
First provide a problem and ask for suggested solutions to solving the problem.
Afterward, guide the students through a few problems.
Then have students attempt problems on their own in pairs or small groups.
Insist they show each step. (you can provide individual group problems and have
them work them out in front of the class or have every group work on the same
problem)
Introduce multiplication and division problems individually, showing step by step
solutions. Depending on students and prior knowledge, this may take several
days.
Check their understanding by offering true or false problems.
Ex: 5 – x = 6 when x = -1 (is this true or false?)
You can increase the difficulty by having them make the equations true or false
themselves.
Be sure to include a variety of rational coefficients and constants to challenge the
students and offer multi-step problems as their confidence increases. Be sure the
pg. 7
Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
students can solve both equations and provide and solve model equations as well.
Ex:
Inequalities (review or teach as necessary), most inequalities are remembered
through some form of mnemonic (alligators or < looks like L for less than, etc).
What’s the teacher doing?
What are the student’s doing?
Leading the activity.
Creating problems based on classroom
progress and increasing or decreasing
difficulty as necessary.
Explaining any confusing concepts as
they arise.
pg. 8
Copying and solving the guided
problems provided showing their work.
Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
Phase Three: Explain the Concept
Day 5: Inequalities
Have the students enter the class picking up (or handing them) a random
inequality index card.
Make sure the students fully understand the symbols and relate to them by using
them in real world scenarios such as: I feel > ok today, I feel < ok today, or I feel
= ok today.
The following two activities can be done individually, in pairs, or small groups and
are especially easy if the students have access to individual computers, smart
phones, or tablets that allow them access to the internet. If access is not handy,
then you can provide prices as needed.
Day 6 – 8
Activity 1: Grocery shopping vs dining out
Computer lab based activity: have students determine whether it’s cheaper to
prepare a meal or to dine out. Based on the classes, you can have predetermined
meals or allow them to create their own.
I would recommend basic classes have a premade menu with prices rather than
researching costs online. They would require more structure and less abstract
concepts.
Advanced classes would be capable of researching costs, field trips to grocery
stores, access to multiple menus and including gas & sales taxes into their
problems.
Activity 2: Vacation trip
Give students a selected budget (random amount as they walk in, predetermined
amount, or have them compete for the amount) to use as they see fit in planning a
vacation trip. (based on the class, this can be highly engaging or might require an
excessive amount of guidance through you)
Guide the students along as necessary (you can even plan trips to a grocery
store). Make sure the students relate the inequalities as they compare them to
their real-world problems.
To encourage all students, provide a rubric showing what you expect. Be sure to
have the students explain the answers they chose and why.
pg. 9
Unit 3: One- Variable Equations, Inequalities, and their Applications
What’s the teacher doing?
What are the students doing?
Is it cheaper to dine in or out?
Answers will vary based on meals.
8th Grade
Providing guidance and encouragement Researching and creating their own
for major activity.
real-world problems
Providing limitations and rules as
needed to prevent possible problems
from arising.
Phase Four: Elaborate on the
Concept
Day 9 - 11
Activities:
Elaboration: Question and answer session... circle format? Can pass a paddle,
toss a ball, multiple ways to encourage random participation as you ask the
students more in-depth involved questions.
Explain inequalities... what are they? Have the students explain what they think of
when you say the word inequality. Have them explain how they would go about
solving an inequality (solved the same way as an equation with a variable). Make
sure they know an inequality can have multiple answers.
Taking the prior week’s activities (vacation and dinner), add additional terminology
to their journals introducing financial literacy problems. New material.
Introduce formulas used to determine interest on credit cards, loans and savings
accounts. This will be taught further in a future unit. Keep financial problems
limited to one variable by providing the other variables for the students.
Incorporate the prior activities the students developed into actual real world
scenarios.
If you place the vacation costs on a credit card, how much interest will you accrue
in a year, a month, a week, and a day based on both simple and compound
interest. This can be done simultaneously with the activities or afterward as its
own lesson but will take multiple days to teach both simple and compound interest.
Repeat the process on the dinner, placing it on a credit card or loan from a friend.
Provide multiple opportunities for the students to estimate how much interest
they'll be paying in addition to the actual price before determining the answers.
Have students compare both simple and compound interest and which would be
better in a credit card or savings account.
pg. 10
Unit 3: One- Variable Equations, Inequalities, and their Applications
8th Grade
What’s the teacher doing?
What are the students doing?
Guiding students through real world
scenarios involving personal finance
situations
Recognizing and determining the better
value of simple and compound interest.
Manipulating variables with guidance to
determine the better value through
estimation.
Offer suggestions as they analyze their
individual activities
Phase Five: Evaluate students'
Understanding of the Concepts
Day 12 Review
Computer-based activity: Review on IXL (k.10 & k.11) or alternate computer
based programs
IXL will allow students to work several problems for free before requesting a
membership. This can be used as a warmup for students if your campus has no
membership.
Day 13
Activity: Assess students based on IFD
pg. 11
Unit 3: One- Variable Equations, Inequalities, and their Applications
Additional resources to use if available: IXL website (membership required)
https://www.khanacademy.org
What’s the teacher doing?
What are the students doing?
Monitor and assess students on IFD
problems. Reteach as necessary.
Displaying understanding of material
based on performance indicator.
pg. 12
8th Grade