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Linear Inequalities

UNDERSTANDING BY DESIGN

 

Unit Overview

Unit Title: Linear Inequalities

Grade Level: Junior

Subject/Topic Area(s): Algebra II

Designed By: Maeve Goetz

Time Frame: 4 weeks; Block Schedule

School District: San Antonio Independent School District

School: Highlands High School

School Address and Phone:

3118 Elgin Ave.
San Antonio, TX 78210
(210) 333-0421

Brief Summary of Unit (Including curricular context and unit goals):
The students will be introduced to linear inequalities and systems of linear inequalities. The main
understandings that I want to develop in my students are that there are many methods to solving math
problems and that solving the problem is not always enough ( mathematically correct solutions are not
always the best solutions). They will need to analyze the solutions they develop to determine whether or
not their answer is reasonable.
Students will demonstrate their knowledge of linear equalities by solving them algebraically,
graphically, and using a table. Students will compare and contrast linear equations to linear inequalities.
They will explore real-world examples of linear inequalities and discuss when linear inequalities are
necessary to solve certain problems.
Students will apply what they have learned to a project. Each student will open a store of their
choosing. They must decide how many of two products they would like to have at their store based upon
the amount of space each takes up and the profit earned by each product. They will also compare this
simplified problem to what the problem would be like in the real world .

 

Unit: Linear Inequalities
Course: Core III
Stage 1: Desired Results
Understandings
Students will understand that:
There are many methods for solving mathematical problems.
Systems give absolute or optimal solutions for a set of equations or inequalities.
Mathematically correct solutions may not always yield the best solutions.
Essential Questions Knowledge & Skill
When do you use inequalities? When do you not?
How do you decide which method to use to solve a
problem?
When is a “correct” mathematical answer not the best
solution?
§111.33. Algebra II
(3) The student formulates systems of equations and
inequalities from problem situations, uses a variety of
methods to solve them, and analyzes the solutions in
terms of the situations. Following are performance
descriptions.

(A) The student analyzes situations and formulates
systems of equations or inequalities in two or more
unknowns to solve problems.

(B) The student uses algebraic methods, graphs, tables,
or matrices, to solve systems of equations or
inequalities.

(C) For given contexts, the student interprets and
determines the reasonableness of solutions to systems of
equations or inequalities.
 
Stage 2: Assessment Evidence
Performance Task: see attached paper.
Other evidence:
Writing samples to accompany one of their examples of solving a linear inequality (explain what they do in each
step and why).
Quiz over solving linear inequalities.
Quiz over solving systems of linear inequalities.
Cumulative Test.
Stage 3: Learning Activities
(Steps taken to get students to answer Stage 1 questions and complete performance task)
Note: The text referred to here is Contemporary Mathematics in Context: Course Three, 2003 ed.

Day 1-2:
• Ask the students the question: what are inequalities (mathematical or otherwise)?
• What are some examples of inequalities? Graph one of the examples given to show how real-world
ideas and examples can be represented mathematically. Emphasize how you determine which region
will be shaded.
• Demonstrate “coding” for linear inequalities. “Coding” should have been covered with solving linear
equations; explain the extensions of “coding” here. After performing a few examples for the students,
have the students work through a set of 10 inequalities where they are to get “y” by itself.

Day 3:
• Explain how to graph inequalities. Again, work a few examples for the students, then walk around the
room and check for understanding. Give them examples where they must first “solve” for y, before
graphing.
• During the last 5 minutes of class, have the students write an explanation of what they are doing and why
they are doing it as they solve a linear inequality.

Day 4:
• Compare and contrast linear equation and linear inequalities.
What is the solution of a linear equation/inequality?
How many solutions does a linear equation/inequality have?
What does the graph of a linear equation/inequality look like?
What does a linear equation/inequality look like and what do the different numbers and variables
represent?
• Have the students find the solutions of a linear inequality algebraically, using a table, and by analyzing its
graph.

Day 5:
• Quiz students over solving linear inequalities.

Days 6-9:
• Compare and contrast systems of linear equations to systems of linear inequalities.
• What are examples of key words that will help you to identify when you will need to use inequalities to
solve a problem?
• Go through Unit 1 Lesson 4 Investigations 1-2

Day 10:
• Quiz over systems of linear inequalities.

Days 11-12:
• Go through Unit 1 Lesson 4 Investigation 3.

Days 13-16:
• Allow students to work on their projects. Keep track of their progress using the following checklist:

______ Day 13: Introduce the performance task. Read through the instructions and go over the rubric. Make
sure students give the name of their store and information (description, size, and potential profit) of
each product involved .
______ Days 14-15: Inequalities written down and graphs completed.
______ Day 16: Reflections completed.

Days 17-18:
• Students will turn in their projects and begin presenting their “stores” to the rest of the class on Day 17.

Day 19:
• Review systems of linear equations with the students.
• Have students write an answer to: How do you solve a system of linear inequalities and what does the
graphical solution look like?

Day 20:
• Cumulative Test

Linear Inequalities Project!
Due in Class on _____________


You are going to open your own store. However, you need to get a business partner in order to get
enough money to get your store running. To show your potential partner your business qualifications, you will
show her the process you go through when stocking your products. You are currently trying to decide how
much of product A and product B you want to buy. Product A takes up less space than Product B, but it also
yields a smaller profit. Note: Assume that there will be no problem selling any amount of either product.

You will need to name your store and decide what you want to sell. Product A and Product B will be of
your choosing (name and describe each product). You will need to decide on a reasonable amount of space that
each product will take up in your store. You will also need to assign an appropriate amount of profit for each
product that you sell. Given those amounts of profits for each product and keeping in mind that you have 600ft3
to store those products, how much do you want to make each month in profit from those products? Write two
inequalities to represent your constraints and explain what they represent. Graph the inequalities and label all
important points and axes.

For the final portion of this project, I would like you to reflect on the work you’ve done. Why was the
use of a system of linear inequalities necessary for this problem? What do you think about the assumption I
asked you to make (at the end of the first paragraph)? What other factors would influence the person’s decision
to become your business partner? Pretend you are receiving this proposal; would you want to invest in the store
if you were in their place? What other costs will come into play if you were really opening up your own store?
What is your favorite thing about your store or presentation? Is there anything you would have liked to have
done differently?

****You will give a short (2-3 min) presentation of your project after you turn it in.****

Due in Class on: _____________
Turn this paper in with your project.

  Fails to Meet
(F)
Approaching
(C)
Meets
(B)
Exceeds
(A)
Description of
Store and
Products

15%
There is no
description of
the store or
products.
Profits and the
amounts of
space needed
for each
product are
given, but none
of the amounts
are reasonable
 
There is a weak
description of
the store and
products.
Profits and the
amounts of
space needed
for each
product are
given, but none
of the amounts
are reasonable.
 
Description of
store and
products is
weak. Profits
and the
amounts of
space needed
for each
product are
given and at
least two of the
amounts
are
reasonable.
Description of
store and
products is
thorough.
Profits and the
amounts of
space needed
for each
product is given
and each is
reasonable

 
 
Inequalities

35%
Inequalities for
the space
constraints and
the profit are
not written
correctly and
are not
accurately
explained.
Inequalities for
the space
constraints and
the profit are
written
correctly, but
are not
explained
accurately.
Inequalities for
the space
constraints and
the profit are
written
correctly and
are accurately
explained.
Inequalities for
the space
constraints and
the profit are
written
correctly and
are accurately
explained and
elaborated
upon.
 
Graphs

25%
Graph is not
accurate or no
important
points or
elements are
labeled.



 
Graph is
accurate, but
not well drawn .
Very few
important
points and
elements are
labeled.

 
Graph is
accurate and
easy-tounderstand.
Most important
points and
elements are
labeled.


 
Graph is
accurate and
easy-tounderstand.
They look
professional
and are all
important
points and
elements are
labeled.
 
Reflection

25%
Very few or no
questions are
answered.
Writing is not
easy to read.


 
Only around
half the
questions were
answered.
Writing is
mostly easy to
read.
 
Most questions
were answered
and elaborated
upon. Writing
is easy to read.


 
All questions
were answered
and elaborated
upon. Writing
is very easy to
read and
grammatically
correct.




 
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