8th grade mathematics is the last course in which students ...



Integrating the TI-73 Graphing Calculator in 8th Grade Mathematics

by

DOLORES LUCIO-GOMEZ

December 2007

A Project Submitted

In Partial Fulfillment of

The Requirements for Degree of

MASTER OF SCIENCE

The Graduate Mathematics Program

Curriculum Content Option

Department of Mathematics and Statistics

Texas A&M University-Corpus Christi

APPROVED: _______________________________ Date:_____________

Dr. Elaine Young, Chair

___________________________

Dr. Nadina Duran-Hutchings, Member

___________________________

Dr. George Tintera, Member

___________________________

Dr. Blair Sterba-Boatwright Chair

Department of Mathematics and Statistics

Style: __APA___

ABSTRACT

Eighth grade mathematics is the final course where students are not allowed to use calculators on the Texas Achievement Knowledge Skills (TAKS) standardized exam. However, the state is requiring immediate implementation of the use of graphing calculators in 8th grade mathematics classes, even though students will not be allowed to use them on the state exam. Research indicates that simple, non-graphing calculators are used on a regular basis in middle schools, often due to cost. However, graphing calculators are ideal for teaching complex mathematical concepts because of their large screen, graphic capabilities, multi-line display, and ability to explore functions by graphing. Technology is becoming increasingly available in middle schools, and it is vital that middle school teachers’ posses the capabilities to use technology, especially the TI-73 graphing calculators, effectively and efficiently in the classroom. This project created lessons with activities and problems to support the use of graphing calculators in the 8th grade curriculum. They will be carefully selected to provide the most effective impact on comprehension and ability of students using the TI-73 calculators in 8th grade mathematics.

Table of Contents

Introduction……………………………………………………………………………4

Literature Review…………………………………………………………………….5

Methodology………………………………………………………………………….7

Results………………………………………………………………………………..8

Summary……………………………………………………………………………..9

References………………………………………………………………………….10

Appendix…………………………………………………………………………….11

INTRODUCTION

Eighth grade mathematics is the last course where students are not allowed to use calculators on the Texas Achievement Knowledge Skills (TAKS) standardized exam. Beginning with ninth grade, the use of graphing calculators in mathematics courses is allowed on the TAKS test given each April. Recently, during professional development training, educators were notified that the state is requiring immediate implementation of the use of graphing calculators in 8th grade mathematics classes. Therefore teachers will be responsible for teaching the use of calculators, even though students will not be allowed to use them on the TAKS exam.

Immediate concerns include funding to purchase graphing calculators, proper training for teachers, and if teachers would actually comply with or ignore the state mandate. To implement the state mandate, new curricula incorporating the graphing calculators is needed. The objectives of this project are to:

• identify existing curricula (if any) for use of calculators in 8th grade mathematics classes

• design 8th grade mathematics lessons targeting the use of TI-73 graphing calculators

LITERATURE REVIEW

Various mathematical organizations encourage the inclusion of graphing calculators in mathematics classes. The National Council of Teachers of Mathematics (NCTM, 1989) calls for student access to appropriate calculator technology throughout their mathematical studies. The American Mathematical Association of Two-Year Colleges (AMATYC, 1995) promotes extensive use of graphing calculators and emphasizes the inclusion of technology as part of an exploratory learning environment. The Texas Essential Knowledge and Skills (TEKS) for middle school mathematics states that students should use graphing technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics (TEA, 2006).

Simple, non-graphing calculators are used on a regular basis in middle schools due to cost (Margaritis, 2003). However, graphing calculators are relatively low cost technology compared to computers (Pullano, 2000). Although the TI-73 is the state mandated calculator for middle schools, schools often inherit TI-83 calculators from the high schools. Districts want to save money and use the technology resources currently available in their district. In order for the middle schools to be compliant with the state mandate, funding is required to purchase TI-73 calculators for middle schools.

Technology is becoming increasingly available in middle schools, and it is vital that middle school teachers’ possess the capabilities to use technology, especially the TI-73 graphing calculators, effectively and efficiently in the classroom. Graphing calculators are ideal for teaching complex mathematical concepts in middle schools due to their large screen, graphic capabilities, multi-line display, and ability to explore functions by graphing (Margaritis, 2003). Graphing calculators are capable of displaying all three major representations of a single function, in algebraic, graphical and data table representation simultaneously (Pullano, 2000).

Harshbarger and Yocco (1999) conducted an inservice workshop for middle grade mathematics teachers in California on using the graphing calculator. They helped teachers explore projects that used the special features of the TI-73 graphing calculator, and made mathematics fun and interesting while the teachers learned. The goal of the workshop was to offer activities that would assist teachers in learning on how to use the TI-73 properly aligned with pedagogy when introducing technology to their classroom. A 26-page booklet gave the keystrokes used with an emphasis on the features unique to the calculator. The workshop and the booklet provided the teachers with interesting middle school projects that were easy to implement into the curriculum. It was this inservice and booklet that inspired this author to create similar lessons and activities for 8th grade mathematics, aligned with the Texas curriculum and standards.

Ellington (2006) noted that when the graphing calculator is used in instruction, but not in the testing process, students do not help to develop the necessary skills to apply mathematical procedures and formulas. In contrast, when the graphing calculator is included in instruction and allowed in the testing process, it helps the students develop procedural, conceptual and overall mathematical achievement skills. A common perception is that if students use calculators they lose their paper and pencil algorithm skills due to dependence on the calculator. Actually, graphing calculators give students and teachers the opportunity and ability to explore, compare, and investigate concepts in a more comprehensive way than if a simple calculator or no calculator was used (Margaritis, 2003).

METHODOLGY

The project began by researching the curriculum, scope and sequence of 8th grade mathematics in Texas. A search to identify existing curricula (if any) for using TI-73 calculators in 8th grade mathematics classes ensued. Then the author designed lessons with activities and problems to support the use of TI-73 graphing calculators. These activities and problems were carefully selected to provide the most effective impact on comprehension and ability of students using the calculators in 8th grade mathematics. The lessons, activities, and problems are in correlation with the TEKS and the scope and sequence of middle school mathematics.

Lessons, activities, and problems from the project were presented to fellow colleagues in several school districts to evaluate the effectiveness of the activities and problem scenario(s) based on their teaching experience. Colleagues were asked to provide feedback in two areas: first, whether the activities and problems were easy to teach, follow, and implement within the current scope and sequence, and secondly, if the activities and problems provide a deeper understanding of mathematics in an interesting manner. Any and all feedback was used to make modifications to the project.

RESULTS

Some curricula are currently available to teachers for integrating the graphing calculator in the 8th grade mathematics classroom. For instance, Texas Instruments provides activities for the TI-73 calculator, but they vary in content and do not cover all that teachers are required to teach students. There are other published activities one can purchase, but the author was unable to determine how thorough and complete they are for middle school mathematics course due to expense.

The project produced ten different lessons with activities and problems for various topics throughout the eighth grade mathematics course. The lessons will assist 8th grade teachers to incorporate the graphing calculator into different topics within the TEKS guidelines. The lessons further enhance the skills students need for the TAKS examination by engaging the student more with skills used to master a concept.

The completed project will be presented to the author’s district for possible implementation in the upcoming school year. It is the author’s intention to incorporate the project curriculum as part of the current scope and sequence calendar for students entering 8th grade mathematics in the 2008-2009 school year. The calculator-based curriculum materials will be available in different formats such as printed pages, a notebook, online, and/or CD.

SUMMARY

Many educators feel the need to include graphing calculators in the middle school curriculum. Research and experience have provided the basis of a calculator-based curriculum written for the author’s district and neighboring districts. It is the author’s intention for the project to be adopted in the district scope and sequence in the future.

It is anticipated that students will be more successful in their high school mathematics courses following their experience with graphing calculators in 8th grade. Increased self-esteem, greater confidence, and more engagement in mathematics will be reflected by improved student grades and higher state exam scores.

This project is important, especially in the author’s district, because the students fall below the state and national averages for completing Algebra 1 in middle school. It will be presented to the district for adoption to assist the teachers and students in familiarizing themselves with the TI-73 calculator and how to use it in mathematics lessons. In the near future, the district will require middle school students to meet the state standard of completing Algebra 1 in middle school and will require teachers to implement the change effectively.

REFERENCES

American Mathematical Association of Two-Year Colleges. (1995). Crossroads in Mathematics: Standards for Introductory College Mathematics below Calculus. Memphis, TN: Author.

Ellington, A.J. (2006). The effects of non-CAS graphing calculators on student achievements and attitude levels in mathematics: A meta-analysis. School Science and Mathematics, 106(1), 16-26.

Harshbarger, R.J., & Yocco, L.S. (1999, November). Preparing in-service middle grades teachers to use the TI-73. Paper presented at the annual International Conference on Technology in Collegiate Mathematics, San Francisco, CA.

Bennet, J.M., Burger, E.B., Chard, D.J., Jackson, A.L., Kennedy, P.A>, Renfro, F.L., Scheer, J.K., & Waits, B.K. (2007). Mathematics: Course 3. Orlando, FL: Holt, Rinehart, & Winston.

Margaritis, A. (2003). Using graphing calculators in the Montessori middle school classroom. Montessori Life, 15(2), 42-3.

Mesa, V. (2007). Solving problems on functions: Role of the graphing calculator. On Learning Problems in Mathematics, 29(3), 30-54.

National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Author.

Pullano, F.B. (2000). Enhancing students’ graph interpretation abilities through the use of graphing calculators. Unpublished doctoral dissertation, University of Virginia.

Texas Education Agency. (2006). Texas Essential Knowledge and Skills.

Texas Instruments. (2004). Topics in Algebra 1. Dallas, TX: Texas Instruments, Incorporated.

Texas Instruments. (2005). Sample Activities TI-73 Explorer. Dallas, TX: Texas Instruments, Incorporated

Texas Instruments. (2006). TI-SmartView [CD-Rom]. Dallas, TX: Texas Instruments, Incorporated.

.

Appendix

Appendix Table of Contents

Letter to the Teacher……………………………………………………………..13

Activity One: Adding and Subtracting Unlike Fractions………………………14

Activity Two: Ordered Pairs……………………………………………………..18

Activity Three: Graphing on a Coordinate Plane……………………………...22

Activity Four: Probability…………………………………………………………27

Activity Five: Slope Intercept Form……………………………………………..31

Activity Six: Solving Equations with Variables on Both Sides……………….36

Activity Seven: Area of Geometric Shapes……………………………………40

Activity Eight: Measure of Central Tendency………………………………….43

Activity Nine: Variance…………………………………………………………..46

Activity Ten: Displaying Data…………………………………………………...49

Letter to the teacher:

The following ten lessons with activities and problems are designed to enhance the mathematical learning taking place in your classroom. The lessons were selected from topics on the Texas Assessments and Knowledge Skills (TAKS) exam. The Texas Essential Knowledge and Skills (TEKS) for middle school state that students should use graphing technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics. These lessons were created to address this requirement. In no way are these lessons meant to replace the manual calculations you have students perform in your classroom. Instead, they are to assist in checking for understanding of the concepts and skills obtained from your instruction and allow you to meet the state requirements. May you find the lessons easy to incorporate into your calendar; the lessons were created to fit into the scope and sequence for middle school mathematics.

Sincerely,

A fellow colleague

Activity One:

Adding and Subtracting Fractions with Unlike Denominators

Lesson Focus:

Add and subtract fractions with unlike denominators.

Materials:

• TI-73 graphing calculator

• Fraction manipulative

• Handout

• Scrap paper

• Pencil

Objectives:

The students will add fractions with unlike denominators.

The students will fractions with unlike denominators.

TEKS:

8.2B Number, operation, and quantitative reasoning: Use appropriate operations to solve problems involving rational numbers in problem situations.

8.14A Underlying processes & mathematical tools: Identify & apply mathematics to everyday experiences.

8.15A Underlying processes and mathematical tools: Communicate mathematical ideas using language.

Process:

Anticipatory Set:

You can use models to show addition & subtraction of fractions with unlike denominators.

1. Draw a picture to show ¼ + 3/8.

2. Draw a picture to show ¾ - 2/8.

Explain how to add and subtract fractions with unlike denominators.

Instructional Input:

Ask students to list the multiples of two and six. Then ask students to identify the LCM. Show them that there are many common multiples, but there is only one least common multiple.

2: 2,4,6,8,10,12,14,16,18, 20, …

6: 6, 12, 18, 24, 30, 36, …

Modeling:

The teacher is to demonstrate by hand the LCM of two different numbers.

3: 3, 6, 9, 12, 15, 18, 21, …

5: 5, 10, 15, 20, 25, 30, 35, …

The teacher will then demonstrate the calculator key strokes necessary to find the LCM of two numbers.

Turn on the calculator

Press the MATH button

1: LCM ( is highlighted Press ENTER once

Enter the first number followed by the “ , ” button

Then enter the second number

Followed by the ) button and press ENTER.

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The teacher will also demonstrate how to key both fractions into the calculator and obtain the solution using the math function key to convert a decimal answer back into fraction form.

¼ + 3/8

Enter 1 ÷ 4 + 3 ÷ 8

You will see .625 on the calculator screen

Press the F↔ D followed by ENTER.

Your answer of 5/8 appears

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¾ - 2/8

Enter 3 b/c 4► – 2 b/c 8 followed by ENTER

You will see the answer ½ on the right hand of the screen

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Guided Practice:

Explain to the students that finding the lowest common denominator (LCD) of fractions with unlike denominators is the same as finding the LCM of the denominators. The teacher will monitor students’ key strokes on the calculator to obtain the LCD of two different fraction denominators when attempting to add and subtract fractions with unlike denominators.

1. 1/2 + 1/6 2. 7/8 – 3/4 3. 1/6 + 2/7

4. ½ + ¾ 5. 8/9 – 2/3 6. 5/6 – 3/8

The teacher will monitor the two different ways to add and subtract fractions using the key strokes demonstrated earlier.

Independent Practice:

The work sheet will contain at least fifteen independent practice problems for them to do. (see attached)

Closure:

Review both methods of finding common denominators. Remind students fractions can only be added or subtracted when the denominators are the same. Ask students to add (-5/4) and (5/6) using both methods demonstrated in the lesson. Remind students to give answers in simplest form.

Worksheet: ADDING & SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS

Find the LCM using the calculator.

1. ¼ + 5/6

2. 1/ 7 + 5/8

3. 1/6 + 2/3

4. 1/3 + 3/10

5. 3/7 + 1/6

Add using the F↔ D button.

6. 7/12 + ¼

7. 1/15 + 7/9

8. 5/8 + 11/12

9. 7/9 + 5/6

10. 3/5 + ¾

Subtract using the b/c button.

11. 2/3 – ¼

12. 7/9 – 1/6

13. 4/5 – 2/15

14. 5/14 – 3 20

15. 6/7 – 5/12

Activity Two: Ordered Pairs

Lesson Focus:

Write solutions of equations of two variables as an ordered pair.

Materials:

• TI-73 graphing calculator

• Handout

• Scrap paper/graph paper

• Pencil

Objectives:

The student will calculate solutions of equations in two-variables as ordered pairs.

TEKS:

8.2B Number, operation, and quantitative reasoning: Use appropriate operations to solve problems involving rational numbers in problem situations.

8.14A Underlying processes & mathematical tools: Identify & apply mathematics to everyday experiences.

8.15A Underlying processes and mathematical tools: Communicate mathematical ideas using language.

8.4A Patterns, relationships, and algebraic thinking: Generate a different representation of data….

Process:

Anticipatory Set:

A moving van travels 85 miles per hour. Use the equation y = 85x, where x represents the number of hours. How far will the van travel in 3.5 hours?

Instructional Input:

Show students a simple two-variable equation such as x + y = 9. Ask students to supply pairs of numbers whose sum is nine. Organize their answers in a table using one value in each pair for x and the other in the same pair for y. Explain that because there are two variables, each solution is a pair of numbers.

Modeling

The teacher will manually demonstrate how to determine whether given ordered pairs are solutions for the given equation. Y =2x + 4, (0,4), (8,2), (1,1), (2,8). The teacher will then demonstrate the key strokes on the calculator to verify if the ordered pairs are solutions of the function.

First demonstrate by hand

X,Y ordered pair

Y = 2 (X )+ 4 (0,4)

4 = 2(0) + 4 true

2 = 2(8) + 4 false

1 = 2(1) + 4 false

8 = 2(2) + 4 true

Second demonstrate by calculator

Turn on the calculator

Press your Y= button

Key in the function y = 2x + 4

On the line \Y1 = press the 2 button

Followed by the x button + button

And finally the 4 button

Press the 2nd button and the GRAPH button.

A table will appear on the screen

Using the ▲▼ you can scroll up or down the table to see which of the ordered pairs provided are solutions to the function and which are not. If you only press the GRAPH button by itself, then the graph will appear.

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Guided Practice:

The teacher will help the students work five problems manually and five problems using the calculator.

Manually calculate and fill in tables for each of the five problems below.

1. y = 2x -1

2. y = x + 5

3. y = -x -6

4. y = 4x + 7

5. y = x

Now use the calculator on the next five functions and complete the tables and draw our graphs on the coordinate planes (provide students with tables and coordinate grids or have them create their own on graph paper).

1. y = x + 1.4

2. y = x - 4

3. y = -1x

4. y = 3x + 2

5. y = -2x + 3

|x |y |

| | |

| | |

| | |

| | |

| | |

| | |

[pic]

Independent Practice:

Handout with fifteen problems (see attached).

Closure:

Teacher needs to discuss with students that there are an infinite number of ordered pair solutions for functions. Show solutions for the function y = 2x-1. The teacher will need to inform students that the solutions to the function may contain negative values and non–integer values. Show other possible solutions for the same equation.

Worksheet: Functions

Directions: Using the calculator and steps provide with this lesson, complete a table with five ordered pairs and a graph for each of the functions. Once the table is complete, write the solutions in ordered pair form.

1. y = 2x -5

2. y = 4 – x

3. y = 2 + 3x

4. y = 4x + 4

5. y = 2x + 7

6. y = 7 – 5x

7. y = 8x + 3

8. y = 9.75x + 2.5

9. y = 2x

10. y = 3x + 1

11. y = 2x -2

12. y = 1.8x + 32

13. y = 1/4x + 3

14. y = 1.2x

15. y = 5x - 2

Activity Three: Graphing on a Coordinate Plane

Lesson Focus:

Graph points and lines on the coordinate plane.

Materials:

• TI-73 graphing calculator

• Coordinate plane/graph paper

• Handout

• Scrap paper

• Pencil & ruler

Objectives:

The student will graph points and lines on the coordinate plane.

TEKS:

8.4A Patterns, relationships, and algebraic thinking: Generate a different representation of data…..

8.7D Geometry and spatial reasoning: Locate and name points on a coordinate plane…

8.15A Underlying processes and mathematical tools: Communicate mathematical ideas using language.

Process:

Anticipatory Set:

Teacher puts a transparency of the school building on the overhead and asks the students to locate different locations within the school. Restrictions are the student can only move up or right and down or left from one location to another. The teacher is to discuss how the student is to write each location as an order pair notation. (Transparency will vary due to each campus).

Instructional Input:

Have students construct a coordinate plane using graph paper. Explain that each axis is a number line, and show how to properly label each axis and how to number the units on the axes. Show students how to identify the location of several different points by providing the coordinates. Then have them graph points and lines in the coordinate plane.

Modeling

Provide the solutions to the function y = 3x - 2 by manual completion of a table and graphing on the handmade coordinate plane. After a short discussion, demonstrate the key strokes to obtain ordered pairs on the calculator and then have the calculator graph the line for the function y = x + 2.

[pic]

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Turn on the calculator

Press the Y= button

Enter the function 3x – 2

Press the 2nd and GRAPH button to see the table

Press the GRAPH button to see its graph picture.

You are then given a set of ordered pairs and are to determine which point is not a solution to a linear function.

{(2, 4), (4, 5), (3, 4), (5, 6)}

Press the LIST button

Under the L1 - key in all of the x-values

Under the L2 - key in all of the y-values

You will need to press enter after each number

Once all numbers have been keyed in you will

Press the Y= button and highlight the PLOT1 and press ENTER.

Press GRAPH and your points will appear

Press 2nd STAT ►►►5

On your screen you will read LinReg (ax+b) Press ENTER

The function y = ax + b a= ? b = ? will appear.

Go back to the Y= button and key in the function and hit Graph

The ordered pair not found on that line is not a solution to that function.

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Guided Practice:

For each set of ordered pairs below, students are to graph each of the ordered pair points and determine which point is not a solution of the a function (the point not on the line with the other three points is not a solution). Have them check through algebraic calculation.

1. {(2,5), (3,7), (4,9), (5,10)}

2. {(1,2),(2,3),(4,6),(6,7)}

3. {(-3,-9), (-1,-5), (0,-3), (2, -1)}

4. {(1,1),(2,2),(-3,-3),(-5,-4)}

5. {(-3,-3),(0,-6),(3,-9),(4,10)}

Independent Practice:

Handout will consist of three problems to be done by hand and six to be done on the calculator (see attached).

Closure:

Draw an unlabeled coordinate plane on the overhead and plot a point. Ask students to identify and label all important parts of the graph.

Inform students that when graphing points on a coordinate plane, the order of the numbers is important. Show them that the points (5, 3) and (3, 5) are located in different places.

Worksheet: GRAPHING ON THE COORDINATE PLANE

Directions: The first three problems are to be done by hand. Construct a table and coordinate plane. Plot the points on the grid and verify which of the four ordered pairs is not a solution to the function.

1. y = -5x + 4 {(-2,14),(0,5),(2,-6),(4,-16)}

2. y = 7 – 3x {(0,7),(1,-5),(-1,18),(2,-17)}

3. y = 2x + 10 {(-4,2),(-3,4),(0,8),(1,12)}

The following three problems are to be keyed into the calculator to verify which of the given ordered pairs is not a solution to the function. Verify that your graph is the same as that of the calculator.

4. y = -5x + 4 {(-2,14),(0,5),(2,-6),(4,-16)}

5. y = 7 – 3x {(0,7),(1,-5),(-1,18),(2,-17)}

6. y = 2x + 10 {(-4,2),(-3,4),(0,8),(1,12)}

The last three problems of ordered pairs are to be completed with the steps shown on the calculator.

7. {(-4,-13),(-2,-7),(-1,-4),(1,-2)}

8. {(-1,-6),(2,6),(3,9),(4,14)}

9. {(0,-1),(1,-.5),(2,0),(3,1)}

Activity Four: Probability

Lesson Focus:

Find the probability of an event by using the definition of probability.

Materials:

• TI-73 graphing calculator

• Plastic bucket

• Marbles of different colors

• Pencil

Objectives:

The student will find the probability of an event by using the definition of probability.

TEKS:

8.2C Number, operation, and quantitative reasoning: evaluate a solution for…..

8.14A Underlying processes & mathematical tools: Identify & apply mathematics to everyday experiences.

8.14B use a problem-solving model….

8.15A Underlying processes and Mathematical tools: communicate mathematical ideas using language.

Process:

Anticipatory Set:

Ask students to count the number of students in the classroom. Ask them if you put their names in the plastic bowl and pulled one out, what is the probability it would be their name? Then ask the students, what is the probability that someone else’s name would be chosen?

Instructional Input:

The teacher will place 20 different color marbles into the plastic bowl (4 green, 5 red, 7 yellow, 4 blue). Shake the bowl to mix up the marbles.

Randomly pick one marble.

What is the first marble more likely to be?

If you draw one marble which color is it least likely to be?

Modeling:

The teacher shows the class that she has one dice. She begins to roll the dice and asks students to predict the probability of the dice resulting in a six on the first roll.

After a short time of rolling the dice by hand, the teacher will ask the students to do it on the calculator by demonstrating the key strokes necessary to do this task.

The following portion needs the BLUE words

Turn on your calculators

Press the MATH button

Right arrow twice then enter 7

On your screen you will see dice (the first number will represent how many rolls we will do.

Place a comma after that number; the second number keyed in is the number of dice we have)

After we close the parenthesis press ENTER. Our six rolls will appear. Results will vary.

[pic] [pic]

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Guided Practice:

Now as a class each will conduct the rolling of the dice with three rolls, five rolls, and eight rolls. Results will vary. Have the students write their results in a table to compare with one another how results of probability vary from person to person and have the students discuss what may be causing the variations.

Independent Practice:

Handout (see attached) Note: when adding more dice to the number of rolls the solutions are the sum of each dice rolled that turn.

Closure:

Inform the students that you are ready to roll a number cube numbered one through six. Ask them to complete the following statements with a vocabulary word.

3. Mrs. ________rolling the number cube is a(n) _____________.

4. Rolling a number less than 4 is an example of a(n)__________.

5. The probability of rolling a number less than 7 is____________.

Worksheet: PROBABILITY

[pic]

Directions:

Conduct the experiment by rolling a single dice 30 times and recording the results of rolling a 1, 2, 3, 4, 5, 6 in increments of five rolls each time so that the data will appear on the calculator screen.

Single Dice

|Number on dice |1 |2 |3 |4 |5 |6 |

|Number of Rolls |30 | | | | | |

|Results- write | | | | | | |

|tally marks | | | | | | |

Student is to conduct the experiment of rolling two dice and record the results after 20 rolls adding up to an even number.

Two Dice

Note the results are the sum of the two dice.

|Sum of roll equals|2 |4 |6 |8 |10 |12 |

|Number of Rolls |20 | | | | | |

|Results- write | | | | | | |

|tally marks | | | | | | |

Activity Five: Slope of a Line

Lesson Focus:

Find the slope of a line and use slope to understand and draw graphs.

Materials:

• TI-73 graphing calculator

• Graph paper

• Ruler

• Pencil

Objectives:

The student will find the slope of a line and use slope to understand and draw graphs.

TEKS:

8.4 A Patterns, relationships, and algebraic thinking: generate a different representation of data given another representation of data…

A.6 A Linear functions: develop the concept of slope as rate of change…

Process:

Anticipatory Set:

Draw a linear graph depicting the average speed a skier made during a 400 mile race. Data points are as follows:

Time 0 1 2 3 4 5 6

Distance 0 50 100 150 200 225 300

Calculate the average speed for the first two hours.

What happened during the third and fourth hours?

Calculate the skier’s average speed after three hours.

Instructional Input:

Slope = Vertical Change = change in y

Horizontal Change change in x

The ratio is often called Rise, or “rise over run”

Run

Positive slope Zero slope

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Negative slope Undefined Slope

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Modeling

The teacher will demonstrate how to calculate the slope of two given points (1,6) and (7,2) using the slope intercept formula.

[pic]

(

6 – 2 = 4 = -2

1 – 7 -6 3

On the calculator you will enter the ordered pairs onto the calculator

LIST enter under L1 1 ENTER 7 ENTER

L2 6 ENTER 2 ENTER

Press 2nd STAT►►►5 ENTER

LinReg (ax + b) appears on the screen

Press ENTER

You will see

LinReg

Y = ax + b

a = -.666666667

b = 6.6666667

Press 2nd ENTRY 2nd ENTRY followed by F↔D ENTER

The result will be your slope.

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Guided Practice:

Provide students with graphs of four lines. Have students use the slope formula to determine the slope between each pair of points and to match the points with the appropriate graph.

A (0,2) (3,8) B (0,0) (2,-2) C (1,1) (3,3) D(1,4) (-1,-6)

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Independent Practice:

Handout (see attached).

Closure:

Remind the students that slope means the steepness of a line and is defined by a ratio of vertical change to a horizontal change otherwise referred to as “rise over run.” Have each student draw four coordinate planes , plot and properly label positive, negative, zero, and undefined slopes.

Worksheet: Slope of Linear Equations

Find the slope of the line that passes through each pair of points.

1. (2,5) (6,7)

2. (4,3) (7, 14)

3. (0,9) (-8,5)

4. (-7,-4) (-3,16)

5. (1,2) (3,6)

6. (-7,0) (-8,10)

7. (2,8) (6,2)

8. (1,1) (3,3)

9. (5,6) (6,7)

10. (3,5) (5,7)

11. Calculate the slope of each graph using the rise over run rule on each picture.

11. [pic] 12. [pic]

13. [pic] 14. [pic]

Activity Six: Solving Equations with Variables on Both Sides

Lesson Focus:

Solve equations with variables on both sides of the equal sign.

Materials:

• TI-73 graphing calculator

• Pencil

• Handout

Objectives:

The student will solve equations with variables on both sides of the equal sign.

TEKS:

8.2 Number, operation, and quantitative reasoning:

(A) select appropriate operations…

(B) use appropriate operations…

8.5 Patterns, relationships, and algebraic thinking

(A) predict, find, and justify solutions to application problems…

8.14 Underlying processes and mathematical tools

(A) identify and apply mathematics to everyday experiences…

8.15 underlying processes and mathematical tools

(A) communicate mathematical ideas using language…

Process:

Anticipatory Set:

We have already learned how to remove parentheses, combine like terms, and even solve simple algebra problems. We will now use these skills to solve problems that involve three or more steps.

Instructional Input:

Let’s study the following multi-step problem.

3(x + 6) = 5x – 2

Step 1 we use the distributive property to remove parenthesis.

3x + 18 = 5x – 2

Step 2 subtracts 5x from both sides to move the terms to the left side of the equation that have variables.

3x + 18 = 5x -2

-5x -5x

-2x + 18 = -2

Step 3 subtracts 18 from both sides to move the integers to the right side of the equation.

-2x + 18 = -2

-18 -18

-2x = -20

Step 4 divides both sides by -2 to solve for x.

-2x = -20

-2 -2

x = 10

Modeling

Now we will solve the same equation using the calculator.

Turn on the calculator

Press the 2nd and WINDOW button

Make sure to set the TblStart = 1

Tbl = .5 as seen on the screen

[pic]

Press the Y= button and key in the left side of the equation under y1 = and the right side of the equation under y2 = as seen on the screen

[pic]

Press the 2nd and GRAPH buttons to see the table. Scroll up or down until you see both y1 and y2 equal each other.

[pic]

When both y1 and y2 equal each other that x value is the solution to the equation.

Guided Practice:

The teacher will work through three problems with the students demonstrating the steps shown above and ask questions to check for understanding and comprehension of how to complete the steps to arrive at a solution.

1. 4 + x – 2 = 8

2. 3c – 10 = 4 – 4c

3. 4a + 4 = 3a – 4

Independent Practice:

Handout (see attached).

Closure:

Review the process for solving multi-step equations

1. Do operations

2. Combine like terms

3. Get variables on the same side of the = sign

4. Undo addition and subtraction

5. Undo multiplication and division

6. Create a poster of these manual steps

7. Create a poster of calculator steps

Worksheet: Solving Equations with Variables on Both Sides

Directions : Using the TI-73 calculator, find the solutions to the equations using the Table Setup feature. Write your answers in the blanks provided.

1. 7 – 10b = -9b + 9 1.______________

2. –x + (-1) = -10 + x 2.______________

3. 3a + 12 = 2a 3.______________

4. 4s + 12 = -2s – 6 4.______________

5. 7 – 5h = 10 – 6h 5.______________

6. 9x – 5 = 8 – 7x 6.______________

7. 7x + 5 = 4x – 10 7.______________

8. 2c + 8 = 10 - c 8.______________

9. y – 3 = y + 6 9.______________

10. 5 – 11w = 13w + 1 10._____________

11. -4g + 3 = 7g - 8 11._____________

12. 6x – 15 = 8 + 2x 12._____________

13. -4x + 64 = 3x - 6 13._____________

14. 3a + 5 = -6 + 2a 14._____________

15. 5m – 2 = m – 10 15._____________

Activity Seven: Area of Geometric Shapes

Lesson Focus:

Find the area of rectangles, parallelogram, triangles, trapezoids, and circles.

Materials:

• TI-73 graphing calculator

• Geometric shapes manipulatives

• Pencil

• Formula sheet

Objectives:

The student will find the area of rectangles, parallelogram, triangles, trapezoids, and circles.

TEKS:

8.7D Geometry and Spatial reasoning: locate and name points on a coordinate plane using ordered pairs of rational numbers.

8.15A Underlying processes and mathematical tools: communicate mathematical ideas using language, efficient tools, appropriate units, and graphical numerical, physical, or algebraic mathematical models.

8.1C Number, operation, and quantitative reasoning: approximate … the value of irrational numbers…

8.2B Number, operation, and quantitative reasoning: ….solve problems….

Process:

Anticipatory Set:

Area = bh Area = bh Area = ½ bh

Area = bh Area = ½ h(b1 + b2) Area = πr²

Vocabulary: diameter, circle, radius, triangle, base, height, trapezoid, square, rectangle, area.

Instructional Input:

Explain to students that area is the number of unit squares inside the figure. For this reason, area is measured in square units. Introduce the formulas for the area of each figure: square, rectangle, triangle, parallelogram, trapezoid, and circle.

Modeling

Manually calculate the area of each of the shapes demonstrating how to substitute the numbers of each figure into the formulas that coincide with each shape.

Guided Practice:

By hand:

Calculate the area of a square with a side measuring 5 cm.

Calculate the area of a parallelogram with sides measuring 8 in. and 10 in.

Calculate the area of a rectangle with measures of 4 x 7 inches.

Calculate the area of a trapezoid with a height of 2 with bases 3 and 7 cm.

Calculate the area of a circle with a diameter of 6 cm.

Students need to familiarize themselves with the formulas in order to complete the quiz on the TI-73 calculator.

Turn on the calculator

Press the APPS button

Scroll down to 2: AreaForm

Press ENTER - 3 times

1: DEFINITIONS AND FORMULAS

Go through each of the shapes listed

The first screen provides a picture with definition of the shape

Area button provided you the formula with a pictorial example

Example button provides an actual calculation of the area of each shape.

[pic] [pic]

[pic] [pic]

[pic] [pic]

Independent Practice:

Students are to take the area quiz level 1 first, upon completing the first quiz students are to take level 2 quiz on the TI-73 calculator.

The calculator keeps track of the number of questions correct.

Press the APPS button

2: AreaFrom ENTER 3 times

2: Area Quiz

Select level and begin to take quiz.

Closure:

Review the vocabulary terms and the formulas from the lesson. Stress the importance of pi, radius, diameter, area of triangles and rectangles when referring to a trapezoid.

Activity Eight: Measure of Central Tendency

Lesson Focus:

Find appropriate measures of central tendency.

Materials:

• TI-73 graphing calculator

• Magazines

• Newspapers

• Pencil

Objectives:

The student will find the appropriate measures of central tendency.

TEKS:

8.12A Probability and Statistics: select the appropriate measure of central tendency to describe a set of data for a particular purpose.

8.14A Underlying processes and mathematical tools: identify and apply mathematics to everyday experiences, to activities in and outside of school…

8.15A Underlying processes and mathematical tools: communicate mathematical ideas using language…

Process:

Anticipatory Set:

Ask students, how teachers decide what grade to give each student at the end of a grading period. Explain that final grades are often based on the average of the grades earned during the grading period. Discuss the meaning of average and ask students for some other situations in which averages might be used.

Vocabulary: mean, median, mode, and sum.

Instructional Input:

Ask students in the class for the month they were born. Create a data list on the dry erase board of the students’ responses. Have students manually calculate the sum, mean, median, and mode of the data.

Modeling

Using the same data collected from the class now calculate the sum, mean, median and mode using the calculator.

Turn on the calculator

Press LIST and under L1 enter the data provided by the class.

After entering all the data press 2nd STAT ►► 7:sum(

Follow by pressing 2nd LIST 1: L1 ) ENTER

Your solution will follow

Repeat the steps for mean, median, and mode.

[pic]

[pic] [pic]

[pic] [pic]

Guided Practice:

Gymnasts in a recent competition received the following scores for their performance on the uneven bars: 9.0, 8.3, 8.5, 9.1, 8.2, 9.0, 8.8, 8.3, 9.2, 9.0, and 8.6. What is the sum of the numbers, mean, median, and mode. Monitor the students as they calculate them on the calculator.

Independent Practice:

Have students search through a newspaper or magazine. Ask them to find a set of data, such as prices of houses, high temperatures, or points scored by the players on a sports team. Students can then find the mean, median, and mode of the data set

Closure:

Ask students to define mean, median, and mode in their own words. Discuss which measure of central tendency would be the best indicator of a students’ grade in a class.

Activity Nine: Variability

Lesson Focus:

Find measures of variability.

Materials:

• TI-73 graphing calculator

• Pencil

• Ruler

• Graph paper

Objectives:

The student will calculate measures of variability.

TEKS:

8.4 A Patterns, relationships, and algebraic thinking: generate a different representation of data given another representation of data…

8.12C Probability and statistics: select…an appropriate representation for presenting and displaying relationships among collected data…

Process:

Anticipatory Set:

To introduce students to variability, review the three measures of central tendency with them: mean, median, and mode. Provide each student with his or her quiz or homework scores so far in the class. Remind students that a measure of central tendency tells where the data is centered. Tell students the topic of the new lesson addresses how a data set is spread out.

Instructional Input:

Begin by defining variability, quartile, and box–and-whisker plot.

Modeling

Review with students how to order a set of data. Find the median and range. Use the median to calculate the first and third quartile. Create the box-and-whisker plot (manually). Now model for students how it is done on the calculator. Explain to students that for the calculator the data set does not have to be put in order.

Turn on the calculator

Press the LIST button

Under L1 enter a student’s quiz or homework scores

Press 2nd Y= turn on the Plot1

Use the arrow to select the box-and-whiskers

Press ENTER

Your Xlist: should be L1

Freq: should be 1

Press ZOOM 7: ZoomStat

Use the TRACE button and the arrows to see all five of the statistical values.

[pic] [pic]

[pic] [pic]

[pic] [pic]

Explain how a box-and-whisker plot provides information that is not easily seen by just looking at the numbers.

Guided Practice:

Provide students with the following three examples. Have them calculate the mean, median, and mode of each data set when they create a box-and-whisker plot on the calculator. Monitor their work and ensure students understand the importance of each key stroke, so that they are not just merely following steps.

1. 32, 47, 43, 45 29, 15, 19, 33, 34, 29

2. 88, 78, 74, 56, 82, 68, 66,

3. 8.1, 6.0, 3.4, 6.7, 2.1, 3.2, 5.3

Independent Practice:

Ask the students, How many hours of sleep did each student have the previous night? Have students create two sets of data sets, one for the boys and one for the girls. Have them create the box-and-whisker plot of each data set. Have students draw their solutions on graph paper identifying each of the five statistical values. Discuss any differences between the graph for the boys and girls and the amount of sleep they got.

Closure:

Show students an example of a box-and-whisker plot. Have students identify the minimum and maximum values. Have students identify where the first and third quartiles and the median are located. Ask students which part of the box-and whisker plot represents the range.

Activity Ten: Displaying Data

Lesson Focus:

Display data in bar graphs, histograms, and line graphs.

Materials:

• TI-73 graphing calculator

• Pencil

• Ruler

• Graph paper

Objectives:

The student will display data in bar graphs, histograms, and line graphs.

TEKS:

8.4 A Patterns, relationships, and algebraic thinking: generate a different representation of data given another representation of data…

8.12C Probability and statistics: select…an appropriate representation for presenting and displaying relationships among collected data…

8.5A Patterns, relationships, and algebraic thinking: predict, find, and justify solutions to application problems.

8.14A Identify and apply mathematics to everyday experiences…

Process:

Anticipatory Set:

Ask students to provide examples of graphs and where they have seen them. Discuss the benefits of presenting information in a graph. Ask students to name some types of graphs.

Instructional Input:

Define bar graph, frequency table, histogram, and line graph. Provide visual representations of each type of graph for students to see. Point out that graphs allow you to visually compare data with ease. Discuss the similarities and differences among the types of graphs. Note: each bar in a histogram represents an interval of measurement and a line graph show changes over time.

Modeling

Ask ten students, five boys and five girls, how many books they read in one month. Create a table with this data. Demonstrate on the calculator how to create a bar graph and a line graph.

|Books read |0 |1 |2 |3 |4 |5 |

|boys |2 |3 |3 |1 |0 |1 |

|girls |2 |1 |2 |4 |1 |0 |

Turn on the calculator

Press the LIST button

Under L1 enter number of books read

Under L2 enter the boys’ data

Under L3 enter the girls’ data

Press 2nd Y= turn on the Plot1

Use the arrow to select the double bar graph

Press ENTER

Your CategList: should be L1

Your DataList 1: should be L2

Your DataList 2: should be L3

Vert Hor 1 2 3

Press ZOOM 7: ZoomStat

Use the TRACE button and the arrows to see the number of books read by each group.

[pic] [pic]

[pic] [pic]

[pic]

Repeat the step again this time select the line graph instead of the bar graph.

Press 2nd Y= turn on the Plot1

Use the arrow to select the line graph

Your XList: should be L1

Your YList 1: should be L2

Press 2nd Y= turn on the Plot2

Use the arrow to select the line graph

Your XList: should be L1

Your YList 1: should be L3

Press ZOOM 7: ZoomStat

[pic] [pic]

[pic] [pic]

[pic]

Guided Practice:

Provide students with the following data table:

|Year |CDs sold |DVDs sold |

|1999 |3020 |753 |

|2001 |3078 |1507 |

|2003 |4865 |3001 |

|2005 |5084 |4074 |

Have them create a double-line graph of the given data on the calculator. Monitor their work and ensure students understand the importance of each key stroke, so that they are not just merely following steps.

Independent Practice:

Have students look through their Social Studies book to find a table of values (data such as population or prediction rates). Have students demonstrate their work on the calculator and provide visual representation of their graphs on graph paper and be able to explain their graphs to the class.

Closure:

On the dry erase board create a table to compare and contrast the three types of graphs studied. Ask students to complete the table by listing the characteristics of each graph.

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