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Mathematics Institute (Tougaloo College)

Lesson Plans- Numbers and Operations

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Name Love/Howard/Johnson Date June 06/09/08- 06/13/08 Subject: Algebra II

Grade: 10-12
Cognitive Domain-knowledge, comprehension, and application
Affective Domain-responding and receiving
Psychomotor Domain-imitation, manipulation, precision

Objectives

NUMBER AND OPERATIONS

NCTM Standards

Understand meanings of operations and how they relate to one another.

Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.
Develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices.

Compute fluently and make reasonable estimates.

Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.
Judge the reasonableness of numerical computations and their results.


MDE Mathematics Framework

Understand relationships among numbers and compute fluently. Verify with technology.

1d) Perform computations, including addition, scalar multiplication, multiplication, determinants, and inverses on matrices. (DOK 1)


ISTE Standards

I. Technology Operations and Concepts
Teachers demonstrate a sound understanding of technology operations and concepts. Teachers:

A. demonstrate introductory knowledge, skills, and understanding of concepts related to technology.

II. Planning and Designing Learning Environments and Experiences
Teachers plan and design effective learning environments and experiences supported by technology. Teachers:

A. design developmentally appropriate learning opportunities that apply technology-enhanced instructional strategies to support the diverse needs of learners.

III. Teaching, Learning, and the Curriculum
Teachers implement curriculum plans that include methods and strategies for applying technology to maximize student learning. Teachers:

C. apply technology to develop students’ higher order skills and creativity.

IV. Assessment Evaluation
Teachers apply technology to facilitate a variety of effective assessment and evaluation strategies. Teachers:

B. use technology resources to collect and analyze data, interpret results, and communicate findings to improve instructional practice and maximize student learning.


INTASC

http://www.ccsso.org/intascst.html





























Procedures

Pre-Test
Students will be administered a pre-test pertaining to matrices. (See attached)

Anticipatory set
Students will name real world situations where items are arranged in rows and columns.
Some examples include grocery items, cars in a parking lot, flight arrivals/departures, spreadsheets, desks in a classroom, and city streets.

Teacher Input
The teacher will introduce new terminology including the following:
matrix, element, row, column, determinant, scalar, dimension, inverse matrix, and zero matrix.

While introducing the terms the teacher will design a KWL-chart on the board. The students will create their own charts and complete them.

The teacher will then discuss the terms, allow students to put the definitions into their own words, and record that definition into their notes. They may also draw and label the parts of a matrix.

After the students have a thorough understanding of a matrix, they will be asked to stand and arrange their desks in a three by four rectangular array. They can decide what direction they want their desks to face.

The teacher will give pointers such as "Row across the lake" to aid the students in remembering the difference between rows and columns. The teacher may also tell them to envision a plantation home with big columns standing tall. Students may have input on ways to distinguish between rows and columns.

After the seats are arranged, the teacher will ask for a volunteer to choose any seat and have them to sit in it then ask the class to identify his/her position.

The teacher will choose different students and repeat this action a couple of times until each student can identify their location within a matrix. The teacher can also tell the student where to sit. For example sit in (2, 3) means second row, third column. When done, the students will return to their assigned seats.

Modeling
The teacher will demonstrate how to add and subtract matrices using a visual representation (chart) and the board. Then the teacher will illustrate and compute examples of matrices on the board.

Guided Practice
The teacher will discuss how the dimension of the matrices must match in order to add or subtract elements.

[ A ] = 2 3
4 5

[ B ] = -3 6
4 -1
0 7

[ C ] = -1 4
-6 7

[ D ] = -1 -5
6 10
3 -4

The teacher will work problems on the overhead projector, while the students work along.
Compute the following:
[ A ] + [ C ] [ B ] + [ D ] [ A ] + [ B ]
The teacher will explain that this can't be done because the dimension of the matrices must match in order to add elements.
The teacher will also explain that the commutative property of addition holds true for matrices.

Then, they will work:
[ A ] - [ C ] [ B ] - [ D ] and show that matrix subtraction is not commutative.

The teacher will have selected matrices already prepared. Using their own personal dry erase boards, students will practice computing matrices of their own. Students will be allowed to use their graphing calculators to check their work.

Independent Practice
The students will use matrices to organize and keep track of numerical information, and will apply addition and subtraction of matrices in a meaningful way.

Students will formulate addition and subtractions matrices of their own and make the necessary computations.

Students will then write a short written report of their matrices and their calculations, using clear notations and labeling what they represent.

Check for Understanding
The teacher will constantly check for understanding from student feedback, facial expressions, observations, and students'work samples.

Closure
Using their math journals, students will list 3 things that they are square about the lesson, 2 things that are circling around their minds about the lesson, and 1 thing that is not clear about the lesson (These questions were taken from the What's Circling Around Worksheet.)

Technology
The students will create a spreadsheet using Excel, which is an applied matrix.

Diversity
The teacher will use differentiated instruction, clinical teaching, pair and share and Communicator boards.

Diversity will be addressed according to the needs of the class.

Re-teaching
The teacher will illustrate two matrices on the board, using a different color for each set of matrices. Then, she/he will demonstrate how to add the matrices. The teacher will then illustrate how to subtract the matrices.

Using dry erase boards, the students will add and subtract matrices.

Enrichment
Students will compute the following types of matrices: addition, scalar multiplication, multiplication, determinants, and inverses of matrices.

Homework
Students will add the following matrices:
[A] -5 4 [B] 2 3
9 7 + 0 -1

[C] -17 -19 + [D] -36 30
-32 -25 21 -14

Students will subtract the following matrices:
[A] 2 8 [B] 7 4
0 6 - 0 3

[C] 12 -23 [D] -28 -36
20 -26 - 37 27

Resources
KWL chart
Matrices chart
What's Circling Around Worksheet
Computers


Materials

Paper

Pencils

Rulers

KWL chart

Dry erase board

Markers

Desks

Graphing calculators

Matrices chart

Overhead projector

Math journal

What's Circling Around Worksheet

Communicator boards

Number cubes

Computers

Microsoft Excel software

Evaluation

Rubric
(See attached)

Post-test
(Same as pre-test)




Name Love/Howard/Johnson Date June 06/09/08- 06/13/08 Subject: Algebra II

Grade: 10-12
Cognitive Domain-knowledge, comprehension, and application
Affective Domain Domain-responding and receiving
Psychomotor Domain- imitation, manipulation, precision

Objectives

NUMBER AND OPERATIONS

NCTM Standards

Understand meanings of operations and how they relate to one another.

Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.
Develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices.

Compute fluently and make reasonable estimates.

Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.
Judge the reasonableness of numerical computations and their results.


MDE Mathematics Framework

Understand relationships among numbers and compute fluently. Verify with technology.

1d) Perform computations, including addition, scalar multiplication, multiplication, determinants, and inverses on matrices. (DOK 1)


ISTE Standards

I. Technology Operations and Concepts
Teachers demonstrate a sound understanding of technology operations and concepts. Teachers:

A. demonstrate introductory knowledge, skills, and understanding of concepts related to technology.

II. Planning and Designing Learning Environments and Experiences
Teachers plan and design effective learning environments and experiences supported by technology. Teachers:

A. design developmentally appropriate learning opportunities that apply technology-enhanced instructional strategies to support the diverse needs of learners.

III. Teaching, Learning, and the Curriculum
Teachers implement curriculum plans that include methods and strategies for applying technology to maximize student learning. Teachers:

C. apply technology to develop students' higher order skills and creativity.

IV. Assessment Evaluation
Teachers apply technology to facilitate a variety of effective assessment and evaluation strategies. Teachers:

B. use technology resources to collect and analyze data, interpret results, and communicate findings to improve instructional practice and maximize student learning.

INTASC

http://www.ccsso.org/intascst.html





























Procedures

Anticipatory set
The teacher will create a real life story problem:

Suppose Joe earned three times as much money as Marty during the summer. Together they earned $210. How much did each person earn?

Volunteers will go to the board and write the steps they took to arrive at the answer. The students who arrived at the answer in another way may show their procedures as well.

Teacher Input
The teacher will give an explanation of what a scalar is and its purpose. In doing so, the teacher will explain what matrix multiplication is by writing a variable (x) outside of a matrix and demonstrating how the scalar multiplies each element within the matrix:

a [c d e f] = [ac ad ae af ]

The teacher will then explain and demonstrate how the scalar (a) is multiplied throughout and is similar to the distributive property. A distributive property chart will be displayed.

The teacher will repeat the process using a scalar variable and numerical elements:
a [-2 4 3 -8] = [-2a 4a 3a -8a]
Students will take notes.

Using magnetic numbers, the students will create their own scalar numerical matrices. When done, they will swap their matrices with their classmates and then compute the matrices.

Modeling
The teacher will illustrate problems on the board taken from the Matrix Activity Sheet. The students can volunteer to go to the board and compute the problems. They will be allowed to use their graphing calculators to check their answers.

Guided Practice
The teacher will create a problem to which the students can relate for better understanding:

My aunts, Betty, Martha and Alice are arguing about their favorite fast food restaurant. They decided that they would each purchase the same number of items from the top three picks and see which one was cheaper.

The teacher will ask the following question: how much is Clarisse going to spend at BunKing?

The teachers will aid the students in computing the matrix multiplication. The multiplication would represent the amount each aunt would spend at each of the three eating places.

Independent Practice
The students will look at the process of multiplying a scalar by a matrix and then a matrix times a matrix.

Using the magnetic numbers and letters, the students will create their own scalar numerical and variable matrices. Upon returning to class, the students will swap their matrices with their classmates and then compute the matrices.

Check for Understanding
The teacher will constantly check for understanding from student feedback, facial expressions, observations, and students'work samples.

Closure
Using their math journals, the students will write and explain the process of multiplying matrices. They will also indicate one real life example where matrices can be applied.

Technology
Students will visit the website, http://www.easycalulation.com/matrix/matrix-multiplication.php/. While on this site, they will use their graphing calculator to compute the necessary operations.

Diversity
The teacher will use differentiated instruction, clinical teaching, pair and share and Communicator boards.

Diversity will be addressed according to the needs of the class.
Re-teaching
The teacher will illustrate two matrices on the board, using a different color for each set of matrices. Then, she/he will demonstrate how to add the matrices. The teacher will then illustrate how to subtract and multiply the matrices.

Using dry erase boards, the students will add, subtract, and multiply matrices.

Students will use numbers cubes to produce the following matrices: 4 X 2 6 X 3 2 X 4
They will roll the cube twice with their left hand. If the rolls produce a 3 and a 6, the students will write 36. They will continue the process. Then, they will do the same with their right hand. The students will then label the columns left hand, right hand, and both.

Enrichment
Students will compute the following types of matrices: addition, scalar multiplication, multiplication, determinants, and inverses of matrices.

Homework
Students will multiply the following matrices:
[A] 7 -5 [B] 4 4
5 -4 -5 7

[C] 10 -6 [D] -3 8
1 12 -9 0

Resources
Distributive property chart
Matrix Activity Sheet


Materials

Dry erase boards

Markers

Distributive property chart

Pencils

Paper

Magnetic numbers

Magnetic letters

Graphing calculators

Matrix Activity Sheet

Math journals

Computers

Communicator boards

Number cubes


Evaluation

Rubric
(See Attached)




Name Love/Howard/Johnson Date June 06/09/08- 06/13/08 Subject: Algebra II

Grade: 10-12
Cognitive Domain-knowledge, comprehension, and application
Affective Domain-responding and receiving
Psychomotor Domain-imitation, manipulation, precision

Objectives

NUMBER AND OPERATIONS

NCTM Standards

Understand meanings of operations and how they relate to one another.

Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities.
Develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices.

Compute fluently and make reasonable estimates.

Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.
Judge the reasonableness of numerical computations and their results.


MDE Mathematics Framework

Understand relationships among numbers and compute fluently. Verify with technology.

1d) Perform computations, including addition, scalar multiplication, multiplication, determinants, and inverses on matrices. (DOK 1)


ISTE Standards

I. Technology Operations and Concepts
Teachers demonstrate a sound understanding of technology operations and concepts. Teachers:

A. demonstrate introductory knowledge, skills, and understanding of concepts related to technology.

II. Planning and Designing Learning Environments and Experiences
Teachers plan and design effective learning environments and experiences supported by technology. Teachers:

A. design developmentally appropriate learning opportunities that apply technology-enhanced instructional strategies to support the diverse needs of learners.

III. Teaching, Learning, and the Curriculum
Teachers implement curriculum plans that include methods and strategies for applying technology to maximize student learning. Teachers:

C. apply technology to develop students’ higher order skills and creativity.

IV. Assessment Evaluation
Teachers apply technology to facilitate a variety of effective assessment and evaluation strategies. Teachers:

B. use technology resources to collect and analyze data, interpret results, and communicate findings to improve instructional practice and maximize student learning.

INTASC

http://www.ccsso.org/intascst.html

















Procedures

Anticipatory set
The teacher will create a real life story problem:

Mr. Ojero is ordering a new car. He can choose from 5 models, 12 exterior colors, and 9 interior colors. How many combinations of models, interiors, and exteriors are possible?

Volunteers will go to the board and write the steps they took to arrive at the answer. The students who arrived at the answer in another way may show their procedures as well.

Teacher Input
The teacher will review scalar multiplication by providing examples on the board. The teacher will demonstrate how to multiply a matrix [A] times a matrix [B].

The teacher will emphasize the importance of the dimensions. The column of the first factor must match the number of rows in the second factor.

The students will use color pencils to differentiate between columns and rows.

The teacher will work examples using varying dimensions of matrices.

The students will compare and contrast adding and multiplying matrices using a 2 X 3 matrix. They will do so by constructing a poster in which they display the similarities and differences between the two. They will illustrate the steps required to solve addition and multiplication matrices. They will also provide examples.

Modeling
Using the overhead projector, the teacher will provide examples of how to multiply matrices. In doing so, she/he will apply what is being taught to the real world.

Guided Practice
The teacher will facilitate as the students work in groups to compete in multiplying matrices. The students will be provided note cards with different problems written on them. The same number of note cards will be placed in each box. There will be a box for each group. The first member of each group must select a card randomly from the box and turn the card over at the same time. Next, the two students will race to compute the problem. Whenever they finish, the next person in their group will draw a card from the box and the cycle will continue until all the students in a group has had a turn and completed their problem. Whichever team finishes all their problems will receive a prize (notebook, pencil, eraser, and pen for each member of the team).

Independent Practice
The students will multiply the following matrices:
[A] -15 -3 [B] -1 7
-6 -12 9 -4

Find the product of XY:
[X] = 5 -1 3
7 5 -8

[Y] = 2 9 -6
-1 4 -2
Students will be allowed to use their graphing calculators.

Check for Understanding
The teacher will constantly check for understanding from student feedback, facial expressions, observations, and students' work samples.

Closure
In their math journals, students will write about how matrices were used during the Civil War.

Technology
Students will visit the website, http://www.ucl.ac.uk/Mathematics/geomath/level2/mat/mat126.html and compute the problems.

Diversity
The teacher will use differentiated instruction, clinical teaching, pair and share and Communicator boards.

Diversity will be addressed according to the needs of the class.

Re-teaching
The teacher will explain that in order to multiply matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

The teacher will then direct the students in the following procedure:
Use two pencils to cover the second row of A and the second column of B so that only the first row of A and the first row of B can be seen. Multiply matching elements and add the products. Continue this process until all the elements are multiplied.

Enrichment
Students will perform computations, including scalar multiplication, multiplication, determinants, and inverses of matrices. Students will apply matrices in different kinds of ways. They will solve systems of equations in three variables.

Homework
Students will multiply the following matrices:
[A] 11 -2 [B] 15 14
35 -4 -1 17

[C] 10 -6 [D] -3 18
11 12 -9 1 0

Find the product of ZY:
[Z] = -6 5 3
1 -8 17

[Y] = 12 -9 -0
13 4 -2


Resources
Computers


Materials

Markers

Paper

Pencils

Color Pencils

Dry erase boards

Poster boards

Rulers

Crayons

Overhead projector

Graphing calculators

Notebooks

Pens

Erasers

Math journals

Computers

Communicator boards


Evaluations

Rubric
(See attached)






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