Title III Learning Experience

Learning Context | Procedure | Instructional/Environmental Modifications | Time Required | Resources | Assessment Plan | Student Work | Reflection

LEARNING CONTEXT

Purpose or Focus of Experience

 The purpose of the LE is to increase problem- solving skills, while using technology, in the context of a real world problem.

Connection to Standards

Standard 3 - Mathematics

Patterns and Functions - Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiantly.

Students:

Represent and analyze functions using verbal descriptions, tables, equations, and graphs.

Translate among the verbal descriptions, tables, equations and graphic forms of functions.

Apply quadratic functions in the solution of problems.

Model real-world situations with the appropriate function

Use graphing calculators to analyze mathematical phenomena.

Modeling/Multiple Representation - Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.

Students:

Represent problem situations symbolically by using graphs.

Manipulate symbolic representations to explore concepts at an abstract level.

Use learning technologies to make and verify geometric conjectures.

Use graphing utilities to create and explore geometric and algebraic models.

Essential Question

 What am I ever going to use this math for?

Content Knowledge: Declarative, Procedural

 Declarative Knowledge: Students will know and/or understand that:

• TI-82 calculators can facilitate the problem-solving process
• How to set up tables, windows, graphing, and quadratic regressions. This knowledge will be obtained by using a teacher prepared worksheet
• The equation of the axis of symmetry
• Have a clear understanding that a square maximizes the area. This knowledge will be discovered.
• That the quadratic regression equation is a representation of a real-life phenomenon.

Procedural knowledge: Students will be able to:

• Given a certain amount of fencing, area can be maximized by changing the shape of the figure made with the fencing. Students will get this knowledge by working in groups and "building" the enclosed areas with string
• Use the trace button to find the maximum/minimum point
• Put collected data into a table on the T-82 calculator
• Set the window represent the data
• Find the linear regression equation using the calculator
• Graph the parabola from a table using the calculator

PROCEDURE
Working in groups, students are given an academic challenge worksheet.

Challenge: You have 36 feet of fencing. You are to make the largest rectangular enclosure to house your barn animals. What dimensions would your enclosure have to have the maximum area for your animals?

Steps 1: Cut 36 inches of string, construct and draw at least three possible rectangular enclosures and find the area of each.

DRAWINGS:

Step 2: Create a table to find the base and height of every possible rectangular enclosure

Step 3: Show how the equation h = 18 - b was derived from the formula 2b + 2h = 36. Why did we use the equation 2b + 2h = 36?

Follow the instructions provided by your teacher to learn how to produce a graph, from a table, on your calculator.

When you have finished, make a conjecture by completing this statement:

If you have a fixed amount of fencing to enclose a rectangle, you can get the maximum area by enclosing a _________________.

Students now work on using TI-82 calculators to graph the data and to find the quadratic regression equation. 

Worksheet #2

Scatter Plots - Quadratic Regression

Check to see if you are in the degree mode:

On

Mode

Degree

To Clear Data From the Lists

Stat

4:Clr List

Press: 2^nd ; 1; , ;2^nd; 2; ,; 2^nd, 3, , (for as many lists as you want to clear)

Press: Enter (Done will appear on the screen)

To Enter New Data

Stat

1: edit

Type in the 1^st x - value

Enter

Type in the 2^nd x-value

Enter

Continue until all of the x-values are recorded

Arrow to the left and up

Enter the y-values (Remember to press enter after each entry

Including the last entry)

Type in the values L1 * L2 into L3 column to find the AREA of each figure.

Set the Window to Fit the Data

Window

(fill in the values):

x-min: Set to 2 less than your minimum x - value (Remember, the x -values are the bases

x-max: Set to 2 more than your maximum x - value

x scl: 1

Follow the steps above for the y - values (Remember, the y -values are the areas.)

To Produce a Graph from data

2^nd

Stat Plot

Plot 1

Enter

On

Highlight 2^nd choice

X - list: Highlight L1

Y - list: Highlight L3

Mark .

Graph (Your graph will appear on the screen.)

To produce the quadratic equation of the data

Stat

Arrow over to Calc

6: QuadReg

Enter (Gives: y = ax^2 + bx +c

a=

b=

c=

y=

VARS

5:Stat

Arrow over to equation

7:RegEQ

Gives equation in the form y = ax^2 + bx +c

This is the end of this Learning Experience which is the precursor to another Learning Experience where the students use the graphing calculators and Photoshop to graph and write quadratic equations of the flow of the water in the water fountains.

INSTRUCTIONAL/ENVIRONMENTAL MODIFICATIONS

 Students work in teams to accomplish the challenge that is presented; communication is encouraged.

String is available for the students to use to "build" their rectangular enclosures. (Tactile)

Teacher walks around the room asking leading questions to encourage thinking.\

An overhead TI - 82 calculator is used so all students can see and be successful with the calculator.

TIME REQUIRED

 This lesson took one eighty-minute period.

RESOURCES

TI-82 calculators

Overhead calculator

Worksheets

String

ASSESSMENT PLAN

This lesson lends itself to an application assessment.

Checklist (pass/ fail)

This will be used on the next lesson: Students will use a digital camera to take a picture of the water fountains in the school. Using Photoshop, they will erase pixels along the path and record the coordinates of each point. Using, their newly learned skills, they will graph the data and produce the quadratic regression equation. (Adapted from Mathematics Teacher)

Students will:

Clear data from lists

Enter new data

Set the window to the parameters of the data

Produce a quadratic regression equation to represent the real-world phenonmena\

Use the trace button to find the roots or solutions

Use the trace button to find the maximum/minimum point

Be able to explain the significance of the roots.

STUDENT WORK
(Include samples of student work showing different levels of performance.)

REFLECTION

The learning experience provided the students with the opportunity to connect a real-life problem with mathematics. Technology enhanced the learning process by providing a visual and algebraic link between the concepts and ideas. 

Many students were unable to understand the problem until they were given the string. The visualization of the rectangular areas was very helpful.

This challenge was adapted from a problem presented at a Principles of Engineering graduate class. The table was adapted from the textbook: Geometry :Bass,L, Rinesmith, H., Johnson,S, Wood,D:; Prentice Hall, Inc Upper Saddle River, NJ 1998 p244

As a follow up, the problem can be enhanced by giving the challenge: Given the same amount of string, what figure would maximize the area if one was able to use the side of a barn as part of the enclosure?