Standards & Performance Indicators- commencement
Exploring Transformations:
On The Computer And Using Transformations On The Computer To Create A Unique Design
Generally, the students learn the material faster and better than the traditional way using graph paper and ruler.
Teacher
This experience was created to integrate computers into the regular Sequential 2 math program and to combine ideas from two disciplines, math and art. All participants were ninth-graders taking Sequential 2 and none of them had previous experience with the software.
MST 3
transformations
learning technologies
investigate transformations
graphing utilities
transformations to functions
MST 7
analyze problems/issues
design solutions
work effectively
gather/process information
present results
Despite the extra work, I have continued to employ the self-teaching concept because I feel strongly that it is the best way for students to learn.
Donna Milgrom
Smithtown Freshman Campus
660 Meadow Road
Smithtown, NY 11787
(516) 366-4710
By far the greatest amount of time must be devoted to the teacher’s own self-preparation.
Phase One—Exploration
In the first part, students explore what happens when geometric figures are transformed on the coordinate plane. The teacher uses the first day to review the basic transformations that students learned in Sequential Math 1. During the next four class sessions in the computer lab, the students work together in pairs, moving geometric shapes in the plane, measuring the coordinates of the original and image figures, making conjectures regarding rules for each transformation, and testing their hypotheses. They then formalize their rules as functions. Throughout the activity, the students direct their own learning using only their knowledge of the program learned earlier in the year and the tutorial provided by the teacher.
The class was given 2 to 3 successive days in the lab during the first activity, and then a discussion of the results obtained to date was held in the regular classroom. Thus, all students were able to get feedback regarding their work; those who were successful would know that they were, and those who were in error could be steered back in the right direction. In the lab itself, the teacher serves as a facilitator, observing each 2-person team and making suggestions and answering questions, thus keeping the lessons focused.
Phase Two—Design
In the second part, students use their new-found knowledge of transformations to create an original artistic design, such as a tessellation, according to a suggested real-life situation. Students must work together effectively, process information, observe common themes, and present their results. It also serves to acquaint them with an assignment such as they might reasonably expect to encounter in the world of work.
Students working in teams of two are given the following scenario and instructed to use the computer to create an original drawing which satisfies the criteria.
Imagine the following situation: You are employed by a design firm which creates designs for vendors who manufacture wallpaper, wrapping paper, tile, and fabric. Your supervisor has assigned you to develop a new design which may be sold to one of these vendors.
The design must contain the following elements:
It must employ at least two types of transformations: line reflections, point reflections, rotations, dilations, and/or translations.
It must use at least two colors.
It can be extended to cover at least 75% of the piece of paper.
It must be relatively easy to reproduce.
In addition, you must write an explanation of how you created your design. This explanation must be clear and easy to read, typed on a word processor, and illustrate your knowledge of transformations. It should be easy to understand so that anyone who reads it can duplicate your design from the instructions.
ASSESSMENT
Phase One—Exploration
Evidence of students’ progress is provided in a number of ways. While they are working in the computer lab, the teacher observes their work, answers questions, and asks other questions,which can point them to further discoveries. The tutorial worksheet requires student responses, and students can also record their discoveries in a script box on the sketchpad workpage. Two or three times during the unit the students report to their regular classroom to discuss, compare, and analyze their findings. Homework from the text is assigned as students complete each of their explorations in each type of transformation. At the conclusion of the unit, each student submits a set of four problems using composition of transformations. These are done using a ruler and graph paper (not the computer) and are scored on a scale of 1 to 10 points, exactly as they are on the Sequential Regents’ exams. This assignment assesses individual learning. Each student does the work on his/her own, and is given a separate rating.
Phase Two—Design
Assess whether the students understand how to apply transformations to achieve artistic effects.
Design Rubric
Projects are evaluated according to the following criteria:
The design:
• was created using at least two different transformations. 10
• can be duplicated repeatedly to cover the surface. 5
• uses at least two colors. 5
• is adaptable to a commercial use. 5
• is fairly simple to reproduce. 5
• is pleasing to the eye. 5
Total: 35 points
The design description:
• can be easily duplicated by the anyone who reads it. 7
• is clear and easy-to-read, with no run-on sentences
and no spelling errors. 5
• is typed on a word processor. 3
Total: 15 points
Project Total: 50 points
Each team of two people is given the same project score. Individuals are assessed according to the assignment described above, and on subsequent tests.
APPENDIX
If you are using Geometer’s Sketchpad 3.0 the following Course 2 Transformations worksheet is a guide created for this learning experience.
One basic shape has been divided using dilations, then rotated and reflected. It is easily reproduced to cover a surface and commercially useful. The three colors are eye-catching, but the repetitive triangles render it less appealing to the eye.
The design uses both rotations and dilations; yellow and green are the colors. The design can easily be translated across the page, and it would make a pleasant tile or textile sample. In addition, the students have incorporated the extra element of curved lines (arcs).
The design is very pleasing to the eye and has nice color contrast. The student has used rotation well, but the translations were done using different directions and distances, resulting in a picture which is not symmetrical. This was probably not intentional, but the unequal white spaces are not only disturbing to view, but render the design difficult to duplicate on the page and therefore not commercially usable. Some minor adjustments, particularly the use of one translation vector, would correct the problems.
This design is deceptively simplistic. The two colored figures are not in fact congruent, but were created by adding and subtracting pieces from congruent squares in the manner of M. C. Escher, then rotated and translated to produce a tessellation. This design is both simple to recreate and infinitely extendible. It is adaptable to many color combinations, making it a useful design for any number fabric items, although a different set of colors would have had more personal appeal, I think.
Two shapes have been created from one basic trapezoid through the use of reflections and translations. The minimal use of two colors keeps the design from looking too cluttered, rendering it pleasant to look at, and useful as a wall tile design. It is easy to recreate and duplicate across the page.
Following is a students’ written explanation of how this design was created.
Create an isosceles trapezoid (shorter base = 4.5cm./10nger base = 7cm./ sides = 2.5)
• Keep reflecting the trapezoids until you have the desired length.
• Reflect the trapezoid over the longer base...then the shorter base...then the longer...
• Reflect the column of trapezoids to create to columns...then three...until you have reached the desired length.
• Delete the longer base of each of the trapezoids. Now you should have overlapping hexagons.
• Color in the diamonds created by the overlapping hexagons. All colors are allowable, but we used yellow and green.
This written description received a score of 15 according to the following criteria:
• easily duplicated by the reader 7
• clear and easy-to-read instructions, with no run-on
sentences or spelling errors 5
• typed on a word processor. 3
Total: 15 points
Standards & Performance Indicators- Intermediate
MST 1
inductive reasoning
solve problems
MST 3
reasoning strategies
conjectures/arguments
conclusions using inductive
simple/compound statements
equivalent forms
ratios/proportions/percents
order relations
estimate/make/use
standard/nonstandard measurements
display/describe/compare
critical judgement
check reasonableness
solve problems
estimate probability
simulation techniques
determine probabilities
Materials:
• rulers and/or Vernier calibrators (one for each pair of students)
• triple beam balance (one for each group of four)
• three 1-pound bags on M & M candies - two plain and one peanut; Note: One bag of the plain candies is used in estimating and then as simple rewards. The second bag is used in calculating the mass.
• a small package of M & M candies
Basically, this unit introduces the students to estimation, measurement (linear and mass), and experimental and theoretical probability using a bag of M&M candies. The students conduct surveys in each class to determine the team’s favorite color and compare their results to the company’s research using various charts and graphs on the computer. They are also informally introduced to ratio, proportion, and percentage.
Maureen Gipp
Commack School District
Candlewood Middle School
1200 Carll’s Straight Path
Dix Hills, NY 11746
(516) 595-2784
The concluding activities focus on statistics and their experiences during the activity. In social studies, their research skills will be augmented by studying the history of chocolate, the production of chocolate, and the Hershey Company. Students will be encouraged to design their own magazine advertisement or create a script for a commercial. Using their imagination they will be given the opportunity to create a candy super hero, report on being a “presidential M&M”, or describe their feelings on being the new blue M&M in “the bag.”
It is expected that the students have some knowledge of measurement, statistics, and simple probability in order to succeed with this learning experience.
In all the activities the teacher is primarily a facilitator. The activities are set up so that the students can work independently or question a member of their cooperative group.
For this project, the accelerated and self-contained students were integrated with the Regents math students. Each group had a minimum of one accelerated math student and one exceptionally bright Regents student. The self-contained students were placed with students who had demonstrated to the teacher their ability to explain themselves well and had patience. Both of these skills are vital for the group to be successful. English as a second language (ESL) students are also part of our team. They also were integrated into the mainstream.
ACTIVITY 1: ESTIMATION AND PREDICTIONS
The students had the opportunity to observe for several days a 1-pound bag of plain M&M’s displayed in a clear container and a 1-pound bag of peanut candies in a solid container. After a brief discussion about how many candies are in the clear container, a discussion of how many candies are in the solid container ensues.
Each student is expected to complete the worksheets. The candies in the containers are used as rewards in class and each student is required to place tally marks on charts set up in the classroom to record the results. When all the candies are given out, the frequency of each color is tallied and recorded.
ACTIVITY 2: SURVEY, TALLY, AND FREQUENCY
A quick discussion, led by the teacher, is held about which color(s) appeared to be the most frequent in the clear container.
A survey of the students, and any adults, regarding their favorite M&M color is then conducted by the teacher. The teacher can have a transparency of the chart, modeling for the class how to deal with the tally marks. These results are then compared to their observation of the clear container.
Questions that can be addressed:
• Are the results similar? How?
• Who was included in the sample surveyed?
• How would this compare to a survey of adults?
• What would be the favorite color of the entire team? Why?
Each student is expected to complete the worksheets.
Note: Activities 3 and 4 can be done independently in science and mathematics or combined in a double science/mathematics period.
ACTIVITY 3: MEASURING M&M CANDIES
In science class (or during a double period) each student will work with a partner to measure the diameter of one M&M candy using a metric ruler or Vernier calibrator. The students repeat the process using 20 candies laid in a straight line. They compute the average and compare it to their first measurement.
The second part of the activity deals with mass measurement and the students use a triple beam balance to compute the mass of one candy and then 20 candies. The students receive 20 candies in a sandwich bag. An average mass is then calculated.
Using this information, the student is expected to calculate how many candies are in a 1-pound bag. Each student will then review their observation estimate and their measurement estimate and write their conclusions.
ACTIVITY 4: STATISTICAL PROBABILITY OF M&M CANDIES
In mathematics class (or as part of a double period) each student is given a small bag of M&M candies. Each student records how many of each color of the candies is in the bag. The students are now informally introduced to writing a ratio. A discussion is led by the teacher to see how many of each color the students had. Since each student’s bag is different, the need for percent is introduced and the students are instructed on how to do the conversion on a calculator.
The second part of the activity is the experiment. Each student places the candy back in the bag or some other container, retrieves one candy, records its color, returns it to the bag and repeats this procedure 20 times.
The final activity requires students to compare their two results and draw conclusions.
ACTIVITY 5: M&M CONCLUSIONS
The concluding worksheets were designed to allow the students to summarize some of the data and to draw some conclusions from this data. The concepts of range, median, mode, and mean are also reinforced in this activity.
M&M BONUS ACTIVITIES:
These are extended activities for the student to explore. A choice is given and students are encouraged to come up with their own ideas. These activities are optional and count as extra credit.
ASSESSMENT
Many group discussions are held as the project evolves. All of the questions are designed to further their critical thinking skills. Since the students sit in cooperative learning groups they are able to further clarify any discussions that take place.
The worksheets themselves are graded by the student, another student, and the teacher according to a scoring rubric. These scores are averaged and a grade is assigned. The grade is counted as a project.
Students are also encouraged to do an extra credit task from bonus activities which are suggested. However, student-generated ideas are readily accepted. New additions this year were designing an M&M pillow and a wood candy dispenser as well as other ideas for surveys.
REFLECTION:
I have found that almost all learners are able to successfully complete these activities to some degree. With the lowest functioning students, a lesson in simple probability might prove useful. These students also require more involvement from the teacher whereas the other students work independently.
Standards & Performance Indicators-Elementary
Math, Monarchs, and Metamorphosis
MST 1
ask why
clarify and compare
interpret observations/measurements
adjust explanations
MST 3
draw conclusions
analyze situations
justify answers
logical reasoning
spatial relationships
multiple representations
explain ideas
recognize/describe/extend
solve for unknown
manipulatives
interpret graphs
two/three dimensional
MST 4
living/nonliving variations
life processes
stages of life cycle
survival behaviors
Merri Jones Earl
Chenango Forks Central School District
John Harshaw Primary School
6 Patch Road
Binghamton, NY 13901
(607) 648-7580
BUTTERFLY LARVA
Nasco, 901 Jamesville Ave., P.O. Box 901, Fort Atkinson, Wisconsin 53538-0901, 30 Butterfly Larvae, Cat. #FB01929M, $36.40
Connecticut Valley Biological Supply Co., Inc., 82 Valley Road, P. 0. Box 326, South Hampton, MA 01073, 30 Larvae, Cat.# AT4851, $38.00
Delta, Dept. CB075, P.O. Box 3000, Nashua, NH 03061-3000, 25 Larvae, Cat.
Students are taught to observe and recognize symmetry as it occurs in the natural world. They learn the mutual relation of parts in respect to position such as: limbs on a tree; veins in a leaf; and whorls in a flower. Being able to recognize patterns is a problem-solving tool which can be applied to many real world situations, even at the first grade level. Students are able to predict outcomes and results.
The purpose of this lesson is to give the students an understanding of the mathematical concepts of symmetry and enhance their spatial sense. The concept will be developed as part of an integrated curriculum that will include the study of insects and, specifically, the metamorphosis of the butterfly. Class discussions (characteristics of an insect); hands-on activities (move tiles to change an asymmetrical figure to a symmetric one); readings (What is a Butterfly?); demonstrations (metamorphosis of a butterfly, and questioning: (What can a symmetrical figure do?); problem solving (How can you change a given asymmetrical figure into a symmetrical one?;, and discovery techniques (use mirrors and pattern blocks for discovery of mirror image concept in symmetry) will be integrated. A chart story based on what they have experienced will enable students to strengthen their reading skills as well as review what they have learned.
Symmetry is a part of the natural world and is a significant concept in a wide range of disciplines. Students may begin to understand elements of it early in their education making this lesson appropriate for grades K, 1, 2, and 3.
Teacher
DAY 1
Students should be given the opportunity to observe the metamorphosis of butterflies with either a picture or, preferably, a living larva. They can look at a variety of types of butterflies pictured in the classroom and in reference books. After discussion of similarities and differences among butterflies, students will be led to the conclusion that the wings on a butterfly are mirror images of one another. The term “symmetry” will be introduced. On the overhead projector, students will manipulate a variety of acetate “wings” to make symmetric pairs.
Optional Video Segment*
To give students a specific responsibility while viewing, the teacher will say, “In this video a girl named Winnie, from Winona, is upset because she doesn’t understand something. I think you can help her. When you think you can, raise your hand.”
Play the tape from the beginning to where Winnie says, “...doesn’t sound nice.” Pause and ask the student with the raised hand what Winnie doesn’t understand. Allow students to express ways to explain “symmetrical.” Resume play until arcade game Symmetrical Polygons appears on screen. Pause and explain that a polygon is a shape with straight sides. As each polygon is presented Mathman and Mr. Glitch appears; pause tape and encourage students to predict if Mathman will chomp because the figure is symmetrical. This action will occur four times, so pause tape each time. Resume play after students predict. Resume play and pause after Winnie’s drawing is “flipped over a line.” Ask, “What can a symmetrical figure do?” (Flip over a line.) Stop tape after Winnie from Winona says, “Symmetry can be beautiful.”
Students will practice placing a line of symmetry (six inch, thin Stick) on shapes formed on the overhead projector using overhead geometric shapes such as the hexagon and squares.
DAY 2
Containers of paper pattern block shapes and four 9” x 12” white sheets of construction paper will be distributed to each group of four students. Each student will create a symmetrical design and will glue the design on his/her white construction paper. After work has been completed, each student will receive a thin, six-inch stick to be glued on the shape to show the line of symmetry.
________
* The optional video portion of the lesson uses the ITV series, Square One TV Math Talk #16, Flip and Fold. The video enhances the lesson on symmetry. There are over 125 instructional tv series that support all the State learning standards. Information about instructional tv programs can be obtained from the Education Services Manager at each of the State’s nine local public television stations.
BUTTERFLIES
Berger, Melvin. A Butterfly is Born. Newbridge Communications, Inc. 1996.
Conklin, Gladys & Lathan, Barbara. I Like Butterflies. Holiday House, USA 1960.
Darby, Gene, What is a Butterfly? Benefic Press, Chicago, IL 1958.
Drew, David. Caterpillar Diary. Rigby, Inc., Crystal Lake, IL 1990.
Gibbons, Gail. Monarch Butterfly. Holiday House, NY 1989.
Mattern, Joanne. Butterflies and Moths. Troll Associates 1993.
Sterling, Dorothy. Caterpillars. Doubleday & Co., Inc. Garden City, NY 1961.
Zoobooks. Butterflies. Volume 7 #9,
DAY 3
Students will sit at desks with one-inch square tiles in small containers and count the tiles to be sure they have 12. After 2 minutes of exploration, direct their attention to the overhead projector. On the overhead, 12 one-inch square tiles will be arranged in an asymmetrical figure. Sheets with that same asymmetrical tile arrangement will be distributed. Students will cover this design with their tiles and be encouraged to explore ways to rearrange the tiles to create symmetrical shapes. Students will share their shapes on the overhead.
The teacher will ask the students to dictate an experience chart story to explain the concept of symmetry.
SYMMETRY
Jonas, Ann. Reflections. Greenwillow, New York, NY 1987.
Jonas, Ann. Round Trip. Greenwillow, New York, NY 1983.
McDermott, Gerald. Arrow to the Sun. Viking Penguin, Inc., NY 1974.
Geometry and Spatial Sense. Curriculum and Evaluation Standards for School Mathematics. Addenda Series. National Council of Teachers of Mathematics, Reston, VA 1993.
INSTRUCTIONAL/ENVIRONMENTAL MODIFICATIONS
Desks are arranged in groups of four, allowing students to observe each other and discuss the procedures. Sharing ideas and understandings accommodates the range of abilities and learning styles. The teacher and/or parent volunteer/aide should be available to assist a student who is confused and unable to participate. To further accommodate the students with special needs, puzzles, enlarged illustrations, and computer programs such as Eduquest on-line rotations are available.
EACH STUDENT WILL NEED:
• Container holding 12, one-inch square tile pieces
• Paper copy of asymmetrical design from General Mathpital (In groups of 4 students)
• 9 x 12 inch white construction paper
• Paper pattern block pieces
• Glue
• One thin stick (or coffee stirrer, straw) about 6 inches long (about the diameter of a toothpick (per student)
ASSESSMENT
The students are observed as they worked with the tiles and a checklist is used to note their individual levels of success. If they were able to reposition the tile pieces to form a symmetrical shape, a check was given. A rubric was designed for evaluating a paper-pattern block design based on the student’s understanding of symmetry.
A worksheet which allowed students to indicate their understanding of metamorphosis was given, collected, reviewed, and evaluated with each student in a conference. Another worksheet distinguishing the differences between a moth and a butterfly was evaluated as above. These worksheets are used to provide feedback.
Students create a booklet illustrating their understandings of the stages of metamorphosis. Self-reflection by students occurs during our daily class meetings.
REFLECTION
Some students could be given additional practice during free time. A homework assignment could help some students.
I found that using the Pattern for Tile Arrangement worksheet as a visual aid to constructing the asymmetrical figure works better than placing the tiles on top of the worksheet. Studies with square tiles could be expanded to the study of area measurement and other geometric shapes in helping to meet other math standards. The study of butterflies and metamorphosis could be extended to the study of other living creatures, their physical characteristics, and their stages of development. Use the concept to develop a unit on bats. Symmetry could also lead to the study of its uses in construction, engineering design, and art.
These lessons reflect the constructivist philosophy. Students are given opportunities to discover basic principles through hands-on activities and observation. The integrated nature of the unit enables them to make connections among the disciplines and incorporate new concepts into their understandings.
Students explore symmetry by manipulating acetate butterfly wings on the overhead.
Students construct a symmetrical design.
Students create a chart story to explain their understandings of the concept of symmetry.
Rubric for Symmetrical Design Made from Paper Pattern Blocks
Outstanding: The student is able to construct a complex design that is symmetrical and can place the line or lines of symmetry accurately.
Good: The student is able to construct a simple symmetrical design and correctly place the line of symmetry.
Fair: The student can construct a design that contains some recognizable elements of symmetry but is asymmetrical.
Poor: The student’s design is asymmetrical with no elements of symmetry.
This student excelled in most performance tasks and had no difficulty creating a symmetrical design. The student said he could create a second line of symmetry if he had another stick. The dotted line is where he showed the teacher it would be placed.
SCORE: Outstanding
The student was unsure, lacked confidence in his own ability. He copied pattern design of a friend but was able to demonstrate to teacher correct placement of the line of symmetry and explained the shape was that of a person: one leg and one arm on each side of the body.
SCORE: Good
Harvest Halloween
Context
First-grade students are asked to work with a partner to solve a problem about buying pumpkins. The problem could be made easier or more difficult by changing the total amount of money the pumpkins cost, or by asking each pair or group of students to find as many solutions as possible.
Objectives
1. to solve a ‘’real” problem that is connected to students’ interests (and is integrated with other curriculum areas, i.e., science, social studies, health)
2. to relate mathematics to things students do outside of school (with their families) and thus to see that mathematics is present and useful (both inside and outside the classroom)
3. to see that a problem can have more than one solution
4. to develop students’ ability to communicate mathematically through writing and/or drawing and to share their thinking (with others) through class discussion
5. (to foster curiosity about mathematics) and to make connections between mathematics and other subject areas.
Materials needed
1. at least 4 pumpkins—1 large, 1 medium, and 2 small (if real pumpkins are not available, you could use paper cut-outs of pumpkins)
2. a variety of mathematic manipulatives and collections for the students to use to “stand for” the pumpkins as they work out their solution(s)
3. paper (white & orange), pencils, crayons/markers
4. large chart paper to record the brainstorming and the multiple solutions during the class discussion.
Procedure
1. Present the following problem to the students:
I went to the pumpkin farm with $5.00 to spend. Big pumpkins were $2.00. Medium pumpkins were $1.00. Small pumpkins were 2 for $1.00. I spent all
of my money. What kind of pumpkins could I buy?
2. Place the pumpkins on a table with a sign showing the prices of each kind of pumpkin. Refer to them as you describe the problem.
3. Explain that students may use any materials in the room, or draw, to help them solve this problem. Brainstorm a list of mathematic manipulatives that they use to stand for the pumpkins as they work on solving this problem.
4. After students have solved the problem, they should write and/or draw to explain how they solved it. Give them an opportunity to share their solution(s) with the class.
5. The students then work in pairs or small groups. The teacher should circulate throughout the room to observe them and to conference where needed.
6. As students finish, they should share their solutions. The teacher should record the various solutions on chart paper so that they can see the multiple solutions and check to see if their solution has been mentioned.
Assessment Techniques
1. Observe and record student participation in class discussion by audio-taping and making written notes during and/or after the lesson regarding:
-ability to explain reasoning
-willingness to take risks
-level of involvement and interest in the assignment
-inventiveness in thinking about how to solve the problem.
2. Observe and record information about the student’s work with a partner by taking brief notes during the work session and talking to groups as they work through the problems regarding:
-grasp of numerical relationships
-ability to represent numerical relationships with words and symbols
-process of working with others
-choice of materials and how they were used to solve the problem
-level of involvement in the process.
3. Review the written work done by the students, making notes on the information gained and using the process of analyzing the work to plan the next steps in the instructional process, in terms of individual students and of the class as a whole.