Discrete Mathematics

COURSE: DISCRETE MATHEMATICS

TOPIC: Coding

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

CT Frameworks:

a) Discrete mathematics is an effective problem solving tool.

NCTM Frameworks:

a) Understand patterns.

b) Solve problems that arise in mathematics and in other contexts.

c) Recognize and apply mathematics in contexts outside of mathematics.

d) Use representations to model and interpret physical, social, and mathematical

phenomena.

Enduring Understandings

Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

Students will understand that:

1. Mathematics is used in coding for efficiency.

2. Patterns exist throughout coding.

1. Why is coding used?

2. How is coding used?

3. What are some of the benefits of coding?

4. Where is coding used?

5. How do patterns work with coding?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

1. The benefits of coding.

2. How to decipher some standard codes use

in everyday situations.

3. How to recognize the patterns in these every

day codes.

Students will be able to…

1. Decipher the common codes such as zip code, UPC code, license, and social security code.

2. State the patterns used in some common codes such as zip code, UPC, and CD coding.

3. Find the error in code or the check number

for that code using zip code and UPC.

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

You have volunteered to help the post office during the Christmas rush. Your job is deciphering the zip codes that cannot be machine read and to record them correctly for the post master. A printed list of your corrections explaining the mistakes and how you rectified it will accompany your list. Use your knowledge of the zip code, binary patterns, and check numbers to accomplish this task.

In addition to tests and quizzes, one or more of the following will be used:

1. Cooperative learning activities.

2. Informal and formal checks: homework, problem of the day, and review worksheets.

3. Analysis of real world problems through

projects.

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

1. Cooperative learning activities.

2. Informal and formal checks: homework, problem of the day, and review worksheets.

3. Projects.

4. Workstations.

COURSE: DESCRIPTIVE GEOMETRY

TOPIC: Three-Dimensional Figures

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

CT Framework:

a) Relationships exist among sides, angles, perimeters, areas and surface areas of solids.

b) Indirect measures of volume for some solids can be found through the use of formulas.

NCTM Standards:

a) Draw and represent three-dimensional geometric figures.

b) Visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections

Enduring Understandings

Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

The student will understand that:

Solids can be classified by the shape of the base they have or the number of bases they have.
Formulas are a useful tool when working with solids.

Name the solids that have 2 bases, 1 base and no bases.
What are area and volume formulas for prisms, cylinders, pyramids, cones and spheres?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

1. That there are three classifications for solids.

2. That there are different formulas for area and volume for each solid.

Students will be able to…

Identify the names of solids that have one base, two bases, and no bases.
Calculate the lateral area, surface area, and volume for each solid.

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student

achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

Continuing the landscaping problem: Find the number of cubic yards of woodchips and crushed stone that must be ordered to complete the gardens with a thickness of 4 inches. Remember that there are 27 cubic feet in one cubic yard. Write down all your formulas and show all of your work.

In addition to test and quizzes, one or more of the following will be used:

1. Cooperative learning activities

2. Graphing calculator activities

3. Informal and formal checks: homework checks, problem of the day, and review worksheets.

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

1. Cooperative learning activities.

2. Graphing calculator activities.

3. Homework and review worksheets.

COURSE: DISCRETE MATHEMATICS

TOPIC: Graph Theory

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

CT Frameworks:

a) Discrete mathematics is an effective problem solving tool.

b) Properties of number are used to develop strategies for computation and estimation

and judging the reasonableness of results.

c) Measurements that are not directly determined can be approximated with some

degree of precision.

d) Real world problems may be analyzed.

NCTM Frameworks:

a) Use geometric models to gain insights into, and answer questions in, other areas

of mathematics.

b) Use vertex-edge graphs to model and solve problems.

c) Formulate questions that can be addressed with data and collect, organize, and

display relevant data to answer them.

d) Organize and consolidate their mathematical thinking through communication.

e) Recognize and apply mathematics in contexts outside of mathematics.

f) Use representations to model and interpret physical, social, and mathematical

phenomena.

Enduring Understandings

Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

Students will understand that:

1. Graph theory is a useful tool for solving real world problems.

2. There are several aspects of graph theory.

1. What is graph theory and how is it applicable to real world situations?

2. What are the different types of situations that can be simulated and solved with graph theory?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

1. How to model problems using graph theory.

2. There are various types of graph theory applications.

3. How to model a project, solve it using graph theory, and present the results.

Students will be able to…

1. Draw graphs, tress, digraphs to model real world situations.

2. Examine Euler, Hamiltonian paths and circuits.

3. Solve tournaments , optimization problems, graph coloring problems, and critical path problems.

4. Present and communication how graph theory is used to solve problems in and outside of the context of mathematics.

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

As a consultant hired to work with the building contractor for the new renovation of Waterford High School you must organize all of the jobs needed to complete the renovation project. You will submit your report to the building construction committee as well as the head contractor for the job and it will include a detailed visual representation with the critical path, earliest-start times, and latest start times labeled for each task. You will gather your data and compile a task table and use your knowledge of graph theory to complete your presentation. You must have a minimum of 15 tasks along with their appropriate task time and prerequisite task accompanied in a table. Your findings are critical to the successful and timely completion of the project.

In addition to test and quizzes, one or more of the following will be used:

1. Cooperative learning activities.

2. Informal and formal checks: homework checks, problem of the day, and review worksheets.

3. Projects modeling real world situations.

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

1. Cooperative learning activities.

2. Projects.

3. Informal and formal checks: homework, problem of the day, and review worksheets.

4. Workstations.

COURSE: DISCRETE MATHEMATICS

TOPIC: Fractals

STAGE 1: IDENTIFY DESIRED RESULTS

Content Standard(s)

Generalizations about what students should know and be able to do

Established Goals:

CT Frameworks:

a) Discrete mathematics is an effective problem solving tool.

NCTM Frameworks:

a) Understand patterns.

b) Recognize and apply mathematics in contexts outside of mathematics.

c) Use representations to model and interpret physical, social, and mathematical

phenomena.

Enduring Understandings

Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)

Essential Questions

Inquiry used to explore generalizations

Students will understand that:

1. Fractals exist in nature.

2. Fractals are patterns.

1. What is a fractal?

2. Where do they occur?

3. How do patterns work with fractals?

Knowledge and Skills

What students are expected to know and be able to do

Students will know…

1. What fractals are.

2. Where fractals occur in nature and in mathematics.

3. How to recognize the patterns in fractals.

4. How to create two and three dimensional fractals.

Students will be able to…

1. Explain what a fractal is.

2. Explain some of the fractals that occur in the real world.

3. Demonstrate a two and three dimensional fractal.

STAGE 2: DETERMINE ACCEPTABLE EVIDENCE

Performance Task(s)

Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)

Other Evidence

Application that is functional in a classroom context only to evaluate student achievement of desired results

The art director in your school has asked for your help. She needs to show how mathematics occurs in art. She needs you to present a project board for the school display case that consists of a representation of art using mathematics. You have decided to demonstrate this using fractals. Using your knowledge of fractals and your ability to construct various two and three dimensional fractals, create a presentation that could be displayed in the school show case for the art department.

In addition to tests and quizzes, one or more of the following will be used:

1. Cooperative learning activities.

2. Informal and formal checks: homework, problem of the day, and review worksheets.

3. Demonstration and analysis of real world situations through projects.

STAGE 3: DEVELOP LEARNING PLAN

Learning Activities:

1. Cooperative learning activities.

2. Informal and formal checks: homework, problem of the day, and review worksheets.

3. Projects.

4. Workstations.