At a Glance

Outline For Thematic Unit


Theme: PATTERNS
Key Concept: To strengthen student's engagement and curiosity by building connections

Seeing how numbers work comes before imagining how they work.

Time Frame: 5WEEKS

The Setting

Celebrate all students’ accomplishments
Provide for formal, informal and self-assessment throughout the theme.
Tap into their insights
Engaging students quickly

Topics
Sorting, Ssymmetry, Numerical Patterns

Materials
Multi Media: Books, videos, games, films

iDeA
Engage students in designing lesson goals and projects

Assessment

A World Of Symmetry

GOAL:
This lesson asks students to observe, react and create symmetrical designs found in their immediate and distant environment and culture.

Students Will

Identify four basic forms of symmetry

Recognize symmetry in everyday objects and life forms

Recognize symmetrical patterns in cultural art

Create symmetrical designs

Standards:

Mathematical Reasoning - Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

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

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

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WEEK ONE- WHAT IS SYMMETRY
INTRODUCTION – The Mathematics of Symmetry

A WHOLE CLASSROOM ACTIVITY
The student cuts out shapes of pattern blocks from a sheet of paper. Folding each in half leads to the discovery that both halves are identical in shape and size. From this observation comes a discussion that leads to a definition for the line of symmetry and the symmetrical form of reflection.

Introduce the students to the four basic forms of symmetry: translation, reflection, rotation and glide.

Visual examples galore by using multiple kinds of media. Heavy exposure to symmetry.

The following site supports as a whole class presentation; examples of each symmetrical form http://forum.swarthmore.edu/geometry/rugs/symmetry/basic.html

Students go through the alphabet to identify letters that have a line of symmetry.

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REFERENCE

http://forum.swarthmore.edu/geometry/rugs/resources/glossary.html

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WEEK TWO - IT'S ALL AROUND US
This week's activities will move the students from the math of symmetry to its application.

OBSERVING SYMMETRY –

For Example: symmetrical shapes and patterns found in nature, patterns on walls and floor coverings, fabrics, and jewelry.

Look through print media for examples of symmetry and identify

GOING ON-LINE
View examples of symmetry in commercial and cultural design and nature.

FOLK ART
http://www.folkart.com/~latitude/hex/hexx.htm

Students can use this site to learn how these people combined symmetry and form to design this folk art.

Abstract Ideas: Interpret the meaning of the colors, shapes and images.


ACTIVITY – RUG SYMMETRY
All around the world floor coverings have the art of symmetrical patterning.

The Navajo People

Students will read a bit about the history of rugs:

http://www.indiantraders.com/aboutart/textiles/navrug1.htm.

To view several examples of Navajo rugs

http://www.americantrails.com/at_rugs.html.

Create a design with geometric shape blocks!

ACTIVITY – SYMMETRY IN NATURE
Many living things have symmetry in their shape and some have color symmetry as well.

Show class the butterfly site, http://butterflywebsite.com/gallery/index.cfm to observe the symmetry of color and shape of butterfly wings.

ASSIGNMENT
The student will look for different images each of animal and plant life for symmetrical shape and/or color. Documenting over time and sharing with the class.

Symmetry by Geometric Figures

Certain geometric figures possess line symmetry.

The figures in the photo are only a sampling of the geometric figures which possess symmetry

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At the end of our week in mass exposure to Symmetry-- we will do a fun little art activity that will exercise the brain. I will be interested in seeing how the students draw and if there is any pattern among the students.

BLIND CONTOUR DRAWING
Grade: Any Grade

When making a blind contour drawing, the eye is not watching the hand as it draws on the paper. Contour drawing is an excellent way to train the eye to draw what it really sees rather than what it thinks it sees.

The first contour drawings you do will look funny! However, with practice, you will find that you will be able to accurately record an image on paper without looking at your hand as it draws!

Materials
Pencils
Paper

Chalk

Blind fold

What You Do
Choose an object to draw.

Pick a point on the object where the eye can begin its slow journey around the contour or edge of the object. Remember, the eye is like a snail, barely crawling as it begins its journey.

When the eye begins to move, so should the hand holding the pencil. At no time should you look at your hand as it draws. Try drawing the entire contour of the object without lifting your pencil form the paper.

"Relax and keep it fun.. practice, practice, practice..."

Symmetry by Letters

Certain letters of the alphabet and words possess line symmetry (such as the samples in the photo).

Notice that some possess vertical line symmetry, some possess horizontal line symmetry, and some possess BOTH vertical and horizontal line symmetry.

Symmetry by Caterpillar

This butterfly caterpillar displays line symmetry.
The pink line is the line of symmetry.

Symmetry by Mosaics

Mosaics and art work often demonstrate the concept of reflections and line symmetry.

This drawing has two lines of symmetry, as shown by the white lines

Symmetry by Flower

Many flowers possess line symmetry.

The biologist's term for line symmetry is "bilateral symmetry."

The white line is the line of symmetry.

Symmetry by Butterfly

Nature displays line symmetry in some of its most beautiful work.

The balanced arrangement of symmetry creates pleasing and attractive forms.

The white line in the photo is the line of symmetry.

SYMMETRY

Line symmetry, or just symmetry, occurs when two halves of a figure mirror each other across a line. The line of symmetry is the line that divides the figure into two mirror images. A simple test to determine if a figure has line symmetry is to fold the figure along the supposed line of symmetry and see if the two halves of the figure coincide.

Another name for the concept of line symmetry is reflection

SORTING

To always begin with a book...

Sorting (Math Counts) by Henry Arthur Pluckrose

This is a wonderful book to use to teach the concept of sorting (why objects are grouped together in a particular way). The simple, colorful photographs depicting everyday objects. Very engaging book!

Students benefit from having lots of experiences sorting, comparing and classifying objects before participating in patterning activities.

Sorting Skills: What Do Children Need to Know?

Students are working towards knowing what is meant by the terms, "sort" or "categorize"

Sorting the same objects in many different ways - e.g. using buttons- sort by size, then by color, then by number of holes, then by shiny or dull, etc.

MATH LESSON PLAN WITH SCIENCE

"Sorting Through Spiders"

Education means developing the mind, not stuffing the memory.

- Anonymous

Grade Level: May be adapted to challenge several grades.

Introduction: Utilize photos and drawings of spiders to analyze and sort the attributes. I will leave it very open-ended for the students to choose the categories from which to sort.

Objective: Introduce children to analyzing and classifying data in different ways.

• 
  
  
  Sorting in general groups



• 
  
  
  Introduction to Dichotomous Keys



• 
  
  
  Assess students' scientific knowledge of spiders

Evaluation: Student observation is the key to assessment as this is a group activity.

Can the students sort the illustrations into logical groups and explain why they did so?
Can students make observations, generalizations, comparisons, and extensions regarding the data collected?

Prior Knowledge: Experience with sorting, graphing and the language connected to both is helpful, but this lesson may be used as an introduction to these concepts.

Learning Advice: Any illustration or object may be used, not just spiders. I will use a curriculum designed from the students' science interests to generate ideas.

Tips: Get to know each student at the beginning of the year by asking children to draw their favorite activity, sport, book character, or themselves.

I prefer to integrate and make learning as meaningful as possible, hence the science connection.

I broke the lesson into one week-- because of depth of understanding I hope to achieve. Depending upon the size of my class, I may choose to only use 10-12 pictures at a time or work in smaller groups.

Dichotomous keys: are tools used by scientists to identify specific names of natural objects, such as trees, ferns, wildflowers or insects. "Dichotomous" means "divided into two parts", so there are always two choices in a step to identification, usually in the form of a positive and a negative. That is, if looking at spiders, I may begin by sorting them into "spiders who are black" and "spiders who are not black". In science, a single object is taken through these steps until an identification is made.

Activity Resources:

http://www.w-w-c.org
http://www.nctm.org
http://www.k8accesscenter.org

PROCESS

Day One:
Introduction and Creation of Materials

1. Ask students to draw a spider and include details they know about them. Preferably, pictures should be larger than life (full page) so others can see them when presented to a group.

2. Have children share and describe their pictures either in small groups or with the whole class.

3. Are the children beginning to notice similarities or differences between other students' work? Prompt them to begin discussing what they notice.

Day Two:
Sorting and Graphing Activities

1. Redistribute or ask children to take spider pictures out of their desks.

2. Gather on carpet in circle, with pictures on floor in front of the students.

3. Lay floor graph or two sorting circles on carpet in center of children. Ask the class: What is a way we can sort/graph our spiders? Children will choose groupings but you may need to help them narrow it down. The more ways the children have of looking at and analyzing the information before them the better their learning will be. Write labels for the groups on the index cards or sentence strips (better yet - have the child who chose the grouping write the label). Each student will then place their picture where they think it is appropriate. Problem solve as a group any discrepancies that arise. Do more groups need to be added? Discuss what they notice when activity is complete. If needed, question them to compare data or ask "What if.." to bring them to a higher level of thinking. Honor different students' ideas by resorting the pictures several times. You may wish to keep the headers that are written to put on the wall for the children to refer to and read.

4. If you began with sorting circles, try laying out the pictures on a floor graph or just sort them in straight lines on the carpet. Children need not only the practice thinking about data, but need experience organizing it in different ways. Reuse the circles to create Venn diagrams, too. Much more than one day could be spent on these activities.

5. At this time, there is a choice in how to continue the lesson depending upon your purpose: a. Go on to Day Three. b. Create KWL chart (a 3 part chart: what we KNOW, what we WANT to know, and what we LEARNED) from the information shown and discussed about spiders. Fill in what we KNEW and what we want to LEARN from this point on. Use this information to begin spider unit. Revisit chart and re-illustrate spiders at end of unit, then key the (hopefully) more detailed spiders out.

6. At the end of the session, put the pictures away once more, or post the bar graph up on a wall. We have a magnetic bulletin chalk board, so we leave it up with magnets for a day or two.

Day Three:

Dichotomous Keys

1. Look at spider illustrations once again. Review some of the attributes that were noticed and discussed the last time they were graphed and sorted. To keep the category or header cards, would be helpful.

2. Choose ONE attribute to focus on. Have children divide themselves into two groups, those that have this attribute and those that do not. Use butcher paper, or a chalk board, to record this. Begin at the center top with a box entitled "Spiders". Draw two diagonal lines in different directions down from it and connect to additional boxes with the "has" and "has not" attributes your class has chosen.

3. Choose a second attribute. Both of the previous two groups will subdivide into two more groups by this attribute (so now there should be four groups). Add the new information to the chart, then continue on and sort out each group as many times as possible.

For example: if we began sorting the pictures by spiders who are "black" and "not black", there should be two groups. Then, the class may decide to group by spiders who are "in webs" and "not in webs". Both the black spiders and not black spiders will divide again and four groups are created: black spiders in webs, black spiders not in webs, not black spiders in webs and not black spiders not in webs.

Can the children sort themselves out so each child is in a group by themselves? Can the children remember their list of attributes they were sorted by?

Extensions: Can a pattern be created with the pictures? Try to have the children find a way to do this without using colors. Use the illustrations at a center for the children to continue exploring ways to sort and graph the pictures. Have commercial dichotomous keys available for the children to use.

Take a "field trip" around your school and identify trees, plants or critters.

Conclusion: The discussions and reasoning of the students will be a wonderful insight into how they really see and understand the concepts of analyzing and sorting data.

NUMERICAL PATTERN

Objectives

Students will
1. understand what the Fibonacci sequence is; and
2. understand how the Fibonacci sequence is expressed in nature.

Materials
The class will need the following:
Paper
Pencils
Copies of Classroom Activity Sheet: Finding Fibonacci Numbers in Nature

Procedures

1. Begin the lesson by discussing the Fibonacci sequence, which was first observed by the Italian mathematician Leonardo Fibonacci in 1202.

2. Work with the class to see whether students can develop the sequence themselves.

3. Write the pattern that has emerged in step 2 on the board: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Discuss what “rule” is being followed to get from one number to the next. Help students understand that to get the next number in the sequence, you have to add the previous two numbers. Explain that this sequence is known as theFibonacci sequence. The term that mathematicians use for the type of rule followed to obtain the numbers in the sequence isalgorithm. As a class, continue the sequence for the next few numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…

4. Tell students that the Fibonacci sequence has intrigued mathematicians for centuries. What’s more, mathematicians have noticed that these numbers appear in many different patterns in nature, often creating the beauty we admire. Tell students that they are going to look for Fibonacci numbers in pictures of objects from nature. Make sure that students understand that they are looking for specific numbers that appear in the sequence, not for the entire sequence.

5. Divide students into groups of three or four. Distribute the Classroom Activity Sheet: Finding Fibonacci Numbers in Nature. Tell students to work together to try to answer the questions on the sheet. Make sure that each student fills out his or her own sheet. For my information, the questions and explanations are listed below. We will work on the first example as a class so students understand what they are looking for.

Flower Petals:
Count the number of petals on each of the flowers. What numbers do you get? Are these Fibonacci numbers?(Lilies and irises have 3 petals, buttercups have 5 petals, and asters and black-eyed Susans have 21 petals; all are Fibonacci numbers.)

Seed Heads:
Each circle on the enlarged illustration represents a seed head. Look closely at the illustration. Do you see how the circles form spirals? Start from the center, which is marked in black. Find a spiral going toward the right. How many seed heads can you count in that spiral? Now find a spiral going toward the left. How many seed heads can you count there? Are they Fibonacci numbers?(The numbers of seed heads vary, but they are all Fibonacci numbers. For example, the spirals at the far edge of the picture going in both directions contain 34 seed heads.)
Cauliflower florets. Locate the center of the head of cauliflower. Count the number of florets that make up a spiral going toward the right. Then count the number of florets that make up a spiral going toward the left. Are the numbers of florets that make up each spiral Fibonacci numbers?(The numbers of florets will vary, but they should all be Fibonacci numbers.)

Pinecone:
Look carefully at the picture of a pinecone. Do you see how the seed cases make spiral shapes? Find as many spirals as possible going in each direction. How many seed cases make up each spiral? Are they all Fibonacci numbers?(The numbers will vary, but they should all be Fibonacci numbers.)

Apple:
How many points do you see on the “star”? Is this a Fibonacci number?(There are five points on the star.)

Ask students where else they see the spiral shape in nature. Would they guess that those spirals are also formed from Fibonacci numbers? Do they find this shape pleasing to the eye? To conclude, discuss other pleasing shapes and patterns in nature, such as those of waves, leaves, and tornadoes. Discuss whether these, too, may have a mathematical basis.

Activity:
Creating the Fibonacci Spiral. Discuss how rectangles with Fibonacci dimensions are used in art and architecture. Use the examples of painter Piet Mondrian, who used three- and five-unit squares in his art; the Egyptians, who used Fibonacci dimensions in the Gaza Great Pyramid; and the Greeks, who used these dimensions in the Parthenon.

Discussion Questions
At first glance, the natural world may appear to be a random mixture of shapes and numbers. On closer inspection, however, we can spot repeating patterns like the Fibonacci numbers. Are humans more apt to perceive some patterns than others? What makes certain patterns more appealing than others?

Evaluation
Evaluate students using the following three-point rubric:

Active participation in classroom discussions; ability to work cooperatively to complete the Classroom Activity Sheet; ability to solve all the problems on the sheet

Some degree of participation in classroom discussions; ability to work somewhat cooperatively to complete the Classroom Activity Sheet; ability to solve three out of five problems on the sheet

Small amount of participation in classroom discussions; attempt to work cooperatively to complete the Classroom Activity Sheet; ability to solve one problem on the sheet

Suggested Readings / Books To Use
Symmetry in Nature (Reading Essentials Discovering & Exploring Science) (Library Binding)by Allyson Valentine Schrier

Life By the Numbers Keith Devlin. John Wiley & Sons, 1998. Written as a companion volume to the PBS series of the same name, this book focuses on the role mathematics plays in everyday life. Each chapter examines a different aspect of the world we live in and how mathematics is involved: patterns appearing in nature, the curve of a baseball, the chance of winning in Las Vegas, the technology of the future. Lots of pictures round out this clear and exciting presentation.

Designing Tessellations: The Secrets of Interlocking Patterns Jinny Beyer. Contemporary Books, 1999. For generations, people have created designs using repeating, interlocking patterns—tessellations. In this slightly oversized, beautifully illustrated book, the author shows how the combination of pattern and symmetry can result in stunning geometric designs.

Vocabulary
algorithm: A step-by-step procedure for solving a problem.
bontanist: A Botanist is someone who studies plants.
primordia form:

Standards
This lesson plan may be used to address the academic standards listed below.

• These standards are drawn from Grade level:


• Subject area: MathematicsStandard


• Understands and applies basic and advanced properties of the concepts of numbers.


• Benchmarks: Uses discrete structures (e.g., finite graphs, matrices, or sequences) to represent and to solve problems.


• Uses basic and advanced procedures while performing the processes of computation.


• Uses recurrence relations (i.e., formulas that express each term as a function of one or more of the previous terms, such as the Fibonacci sequence and the compound interest equation).


• Uses basic and advanced procedures while performing the processes of computation.

GOLDEN

Fibonacci Hunt (Review)

Fibonacci numbers show up often in nature. The Fibonacci sequence is named for Leonardo Fibonacci, an Italian mathematician who lived from 1170-1250. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21… with each term being the sum of the previous two terms (8 = 3+5, 13 = 5+8, 21 = 8+13, etc.).

Give students the first eight terms of the sequence and challenge them to extend the pattern. Ask them where these numbers show up often in ( plants, pinecones, sunflowers, artichokes, celery, lilies and daisies).

Bring in a pineapple and demonstrate the Fibonacci numbers. Looking at the bottom of the fruit, you can easily see spirals in the patterns. A pineapple makes for an excellent demonstration – the pattern of polygons on the husk forms several spirals that can be followed in different directions. Count the number of spirals in different directions (masking tape helps), and you'll find they are Fibonacci numbers. You'll probably find 13 spirals in one direction, eight in another and five in a third direction.

Golden Ratio Sequence

Demonstrate how students can find the ratios between consecutive Fibonacci numbers:
1/1 = 12/1 = 23/2 = 1.55/3 = 1.666...8/5 = 1.613/8 = 1.615384...... Etc.

Point out that the ratios are going up and down but getting closer and closer to some value. That value is about 1.61803…an irrational number that is called "the golden ratio." The number has fascinated mathematicians and artists for millennia.

Now have students create their own "Fibonacci-like" pattern by choosing two starting numbers and applying the same rule to build a sequence. For example, if one chooses 3 and 7 as the starting numbers, the sequence is 3, 7, 10, 17, 27, 44, 71, 115, etc.

Ask your students to find the ratios between the terms of their own sequence. They will be surprised to find that their sequence, and everyone else's in the room, also converges to the golden ratio! Wow!

Golden Rectangle

There are many ways to construct golden rectangles; here's a fun way that uses Fibonacci numbers. Supply students with graph paper. Have them outline a 1 x 1 square near the center of the paper. Add a 1 x 1 square adjacent to the first, then a 2 x 2 square so that it touches both of the 1 x 1 squares. Continue to add squares, circling around the outside of the figure. In each square, students write in the side length – the lengths form the Fibonacci sequence.

Golden Spiral

When the page is filled, students can draw a quarter circles in each square in order to form an approximate "golden spiral" as shown above. This spiral is related to the shape of ram's horns, sea- shells and other spirals found in nature.

This is just a small taste of the wonders of mathematics in nature.

Art Project and Demonstration:

Introduce a book about Patterns and Relationships:

Mathematics is the science of patterns and relationships. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest elaborateness or complexity but in finding the greatest simplicity of representation and proof.

Activity: Homemade spirals.

The photo above shows washers laid out in a string, starting at a center. Each washer touches the previous one, and each wrap around the center just touches the previous wrap. No pattern is obvious at first, but after a number of wraps, a pattern of additional spirals emerges.

The sunflower seed head is an example of botanist William Hofmeister's 1868 observation that primordia form preferentially where the most space is available for them. They also must form where they attach efficiently to the rest of the plant, and this is a geometric consideration. The pattern can also be modified by moisture and nutrient conditions that affect the size of forming seeds.

You don't need biology to produce spirals similar to those found in sunflower seeds. Here hardware store washers have been laid out. After about six or seven wraps additional spiral patterns develop, just as in the sunflower, spiraling steeply outward, curving away from the center. The reason is simple. The growth pattern of the seed head (and our constructed spiral) is such that it is biased to povide reasonably close packing of the seeds (or washers) consistent with the growth processes.

Mathematical thinking often begins with the process of abstraction—that is, noticing a similarity between two or more objects or events.