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Planning a Math Night

As you plan your math night this year, where do you start?

Combining school, parents/guardians and students together to enjoy math is a goal of Math Nights. Math instruction has changed! The "Direct Model" or "Hands On" activities help make math accessible to everyone and Math Night is a good forum to share materials with parents.

Many parents suffered through math and feel that they don't "have the brain" for math. Other parents want to know about the different strands of math that their students are now learning in school. It's not just about computation anymore. Planning a Math Night for Middle School students can be overwhelming, but a worthwhile experience.

Parents will enjoy exploring mathematics with their students using activities that involve concrete materials. As you select materials and activities, keep in mind that parents and students need to be engaged and feel successful.

Math Night Ideas

I recommend using some of these activities from the Summer Math Content Training:

Through the ESD 112 Media Center, you can borrow a Gems Kit for grades 5-8 entitiled Math Around the World. The kit includes the manual and all of the materials needed for the activities. Toward the back of the guide there is a plan for making Math Around the World a school-wide festival. It is a good idea to look over the materials and perhaps teach the games to your students before the Math Night allowing the students to become experts for their parents.

Math Nights are well worth the effort. They help encourage parents/guardians to share experiences with their students with a positive school activity. Refreshments and child care will enhance the evening and are encouraged! Student enthusiasm for time with their parents will make this a great event.

Family Math Resources

The following is a list of Family Math guide books:
(These lessons are good to use in the classroom too.)

Family Math

By: Jean Kerr Stenmark, Virginia Thompson, and Ruth Cossey.
The handbook includes activities for students grades K-6 as well as information on how to organize a family math night. Materials needed are general classroom items such as calculators, beans, dice, graph paper, counters, cubes.

Family Math the Middle School Years

By: Virginia Thompson and Karen Mayfield-Ingram
This manual focuses activities on algebraic reasoning and number sense for grades 5-8 and includes a section on Organizing Family Math in the Middle School. Materials needed are general classroom items such as calculators, beans, dice, graph paper, counters, and cubes.

Family Math II

By Lawrence Hall of Science
A new manual which can be previewed here.

Math Trails

As you look for ways to show students math in their world, you may want to design, or have students design, a math trail.

Math is everywhere. Math trails encourage active participation as participants observe, investigate and problem solve. Students make connections between math in the classroom and their environment.

After choosing a starting point, design questions relying on observations, gathering data, measuring, estimating, and making conjectures. Students record their observations in written form with sketches as well as using a digital camera.

Each group needs directions and a tool kit with string and a tape measure. You might start with a question to encourage them to take a moment and look your surroundings over. What geometrical shapes predominate? Share your reasons. One challenge is designing clues to lead participants to the correct location.

Participants can be students of any age including adults and families. Students could design math trails for their peers or younger students.

Math Trails: One Path for Making Connections by Janet Mock and Jerry Johnson, 2000, provides more information on math trails.

Math Trails in Ottawa has additional information.

Students will become more aware of their environment as they participate and design math trails.

Quilts: Reflection, Transformation and Symmetry

Guest Author: Colleen Bellas

Developed as part of a unit on geometric patterns and designs for middle level mathematics students, this lesson is designed to help students see the importance of patterns in geometric design and in quilts.

Project Scenario:

A square from a quilt was found in the basement of an abandoned house. The new owner needs someone to re-create this possibly historic quilt by using patterns of the square. The owner believes the quilt tells the history of the old house.

You have been asked to recreate the design of the quilt from possibly the late 1800s to the early 1900s using patterns from the one quilt square. From the one square, use transformations and symmetry to create at least a nine patch quilt design. Create a story about the quilt based on historic facts from the time period.

Keep a reflective journal documenting the quilt re-creation process with words and sketches. Include a complete description of the geometric shapes used in the quilt and how the design process was developed. Please include the appropriate geometric vocabulary and transformation descriptors. Word process the journal using MS Word or paper and pen if a computer is not available.

About the project:

Students need to understand how to transform geometric shapes to create patterns by flipping, slipping and reflecting the shapes and the concept of symmetry prior to the lesson. Drawing software such as Microsoft Draw, Tessellation Exploration, and Geometer's Sketchpad will be useful in the lesson.

Students should be divided into two to three member small groups; each group having access to a computer for the entire period for 3-4 class periods. If there are not enough computers, have groups rotate onto the computer; the project will take more class time.

Research the history of quilts to start the project. Look at some of the Web links below to learn about the role of quilts in history:

Choose a quilt square from the Quilt Samples Web link. Decide what time period this quilt might be from and begin brainstorming how your completed quilt might look.

Using software*, re-create your quilt. The quilt needs at least three variations of the one square by transforming the square and including multiple colors. The quilt should also include several squares that have symmetry. (*The software can be MS Draw, Tom Snyder's Tessellation Exploration or Key Curriculum's Geometer's Sketchpad.)

Write a story the quilt tells about the history of the house. Use Inspiration software to brainstorm ideas for the story. Then write and word process a one-page quilt story.

For more information on Patterns and Symmetry see these websites:

This starts the discussion of patterns in everyday objects. Mathematics is all around us!

Literature Connections

In this month's sampler, I will be sharing some children's literature books useful in captivating and encouraging student's curiosity about mathematics.

These two books by Jonas are of high interest to students as they ponder how to design their own pages. The stories are read from front to back and then rotated 180 degrees and read the other way:

Explore tangrams with these books:

Here is a lesson using One Grain of Rice.

These books will help building and extending circle and geometric concepts:

When dealing with functions, use The Sneetches and Other Stories, by Dr. Seuss.

These two books are useful in dealing with logic. Shuffle laminated pages and pass them out to students. Working in a group, their task is to figure out the order the pages go in.

Quilting provides a means to explore the concepts of translation, reflection, rotation and symmetry:

Before high stakes tests inspire your class with Dr. Seuss Hooray for Diffendoofer Day! by Jack Prelutsky and Lane Smith.

Listed below is a sampling of the Geometry topics in G is for Googol, by David Schwartz:
D is for Diamond
E is for Equilateral
O is for Obtuse
R is for Rhombicosidodecahedron
S is for Symmetry
T is for Tesselate

Other books by David Schwartz such as How Much is a Million?

Portland School District's site has a description of these additional books:

Any books about MC Escher's work are of high interest to students. Some of Escher's graphics include optical oddities.

Refer back to Literature Random Sampler for some other ideas of using literature in the classroom.

NIM & Math Night

This game/lesson, based on the game of NIM, would be valuable to do with students or adults in a family math night. Playing in groups of two, groups will need 10 toothpicks for the parachute ropes and a print out of the parachute. Each turn a person removes one or two toothpicks. The last person to pick up a toothpick wins. After playing several times, discuss and list strategies and discoveries.

sv.berkeley.edu/showcase/media/cl_balrd.pdf

Play two more games. Again explore participant discoveries.

This site shares a dialogue about NIM and the strategy:
mathforum.org/library/drmath/view/56779.html

Another variation of NIM, online with fruit:
www.2020tech.com/fruit

If using this with adults, discuss the question, "Where's the math?" It wouldn't hurt to discuss this with students as well.

Something to consider in planning your family math night: many adults are uncomfortable with math. Do different activities than what you did with your students, so your students don't overwhelm the parents by already knowing how to do the activities.

Many adults want to help their students but aren't sure what to do. This site shares challenges for families to work on together:
www.figurethis.org

Here is a link to parent resources:
lhsparent.org

Here is a link to the PTA website with a connection to a brochure, 100 Ways for Parents to Be Involved in Their Child's Education:
www.pta.org/parentinvolvement/standards/pdf/App_E1.pdf

Exploring Circles, Ratios & Pi

Circles are everywhere! Since pi is the ratio of circumference to diameter, students can discover the value of pi for themselves. Use a piece of string to measure and record the circumference and diameter of several round objects. Using a calculator, divide the circumference by the diameter and record the value for each object. Find the average of the answers to the division problem. Repeat the activity using metric measures to determine if the effect on the relationship.

mathforum.org/paths/measurement/disc.pi.html

Have you ever seen a tree big enough to drive a car through? Let students predict the diameter of a tree using the circumference and pi. In another problem ask them to determine what circumference a tree needs to be for a car to drive through it and of course have and the tree remain standing.

www.figurethis.org/challenges/c15/challenge.htm

An important concept for students to understand is pi as a ratio between the circumference and the diameter, calculated by the dividing the circumference by the diameter. Use a dynamic software, such as Geometer's Sketchpad, to explore the relationship of a circle, the diameter and pi as the circle changes size and shape.

Spreadsheets are useful to keep track of the results and calculate the value of pi. Also, students may graph their results with diameter on the x-axis and circumference on the y-axis. Figuring out the slope gives the value of pi.

Use a discussion of pi as a way to help make connections between math and history.

mathforum.org/isaac/problems/pi1.html

This site shows values of pi at different times over the last 4000 years.

mathforum.org/isaac/problems/pi2.html

A day has been set aside to celebrate pi, 3/14. (It is also Albert Einstein's birthday.) Visit the following Websites for ideas to celebrate Pi Day:

Introduce the concept of pi as an irrational number to stretch their thinking. Irrational numbers are non-repeating, non-terminating decimals and can not be represented by a ratio of two integers.

Hint for drawing a circle: Tie two ends of a string together to form a loop. Hold one end of the loop in place, while putting a pencil in the other end draw the circle. This method is useful to draw large circles.

Moiré

Use moirés as a way to capture your students' attention. In wallpaper or fabric, the moirés have a shimmering appearance, seeming to move and vibrate. Repetition of simple forms and colors create vibrating effects and moiré patterns. Sometimes the effect is an exaggerated sense of depth, foreground-background confusion, as well as other visual effects.

Moving one pattern over another can generate fascinating visual effects. Students can create their own moirés by overlapping materials with repetitive lines. Copy two identical transparencies and move one across the other on an overhead.

Moirés, sometimes an unintentional form of optical illusions, are an interesting phenomena for students to experiment with in the area of movement, optics, and illusions. This website links to some samples moirés and some experiments to try:

www.exploratorium.edu/snacks/moire_patterns.html

Some people are frustrated by moiré patterns. This links you to moiré patterns with fractals and animated circles to dazzle your students:

www.permadi.com/java/moire/pattern250.html

Several examples on this site will capture your students' attention:

www.sandlotscience.com/Moire/Moire_frm.htm

Here is a link to samples to share with your students. The effect is enhanced by moving parallel lines:

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/moire/moire.html

The effect can be eye-straining. Some people feel moirés contaminate the graphic and have no place in graphical design.

Some artists use tricks of visual perception and perspective to create illusions. Victor Vasarely and M.C. Escher are two artists who experimented with various forms of visual tricks and paradoxes. For more information on these artists see the June 2002 Random Sampler:

edtech.esd112.org/no_limit/rs_archive.html#art

Optical art is a possibility for further exploration of moirés:

www.hhmi.org/senses/a110.html

Tangrams

Tangrams are useful in modeling fractions, developing algebraic sense and geometry concepts, and building math vocabulary. Tangrams originated in China and include seven pieces made up of five triangles, a square and a parallelogram placed together to create designs. An individual piece is called a tan.

This website includes directions for students to make their own tangrams from a rectangular piece of paper:

forum.swarthmore.edu/trscavo/tangrams.html

Have students make their own tangram sets, or if you have the tangram pieces on the Ellison dye cutter, cut the tangram pieces out of cardboard cereal boxes. Wax paper and scrapes of laminate work nicely on the overhead if you don't have overhead tangram pieces.

Use the tangram pieces to create polygons. Using 1 piece, 2 pieces, 3 pieces up to 7 pieces to make different polygons, students organize and label their findings by the number of sides. The chart should include triangle, rectangle, rhombus, square, trapezoid, pentagon, and a parallelogram.

Another tangram lesson uses tangram pieces labeled from A-G. Students write equations using the math symbols and letters to record equations. The activity also provides students the opportunity to draw and label fractions and move to algebraic equations. Students have opportunity to describe findings as well as drawing and labeling solutions. This lesson is a good extension or follow-up of the Build It with Geoblocks lesson.

An inexpensive manipulative, the tangram provides students with tactile experiences and is rich in literature connections such as Grandfather Tang's Story by Ann Tompert. As an extension students can reassemble pieces into a square.

Geoboards

The geoboard is a useful tool in teaching and assessing measurement, working with fractions area and perimeter and two-dimensional geometry concepts.

Problem solving questions can range from:

Slide, flips and turns can easily be investigated.

Before you start the lesson you may want to give the students free exploration time, especially if the students haven't used geoboards before. Begin by letting students freely make whatever shapes or designs they wish.

Encourage students as they complete to show another way to solve the problem or to find multiple solutions.

It is important to have students record and label their discoveries. Geoboards recording sheets are in the back of Van de Walle's book Elementary and Middle School Mathematics, Teaching Developmentally in the blackline master section, #37 or online:

mathforum.org/trscavo/geoboards

Use these phrases to give students the opportunity to journal; reflecting on their thinking.

The teacher's role when students work on a geoboard problem is to walk around the room, listening and observing students. Ask students open-ended question and encourage conversation between students about the mathematics.

Clear geoboards work nicely for student to show their work on an overhead or document cameras. Since the geoboards are transparent, other students also see their work when it is held in the air to show the teacher. Overhead pens can be used to label sections or write vocabulary on the geoboard. Also, if students are having difficulty seeing two shapes as congruent, building it on another clear geobard and placing it on top of the original may clear up the confusion.

When using manipulatives, all students have access to the lesson. A teacher quickly can see and extend the activity if it is too easy and adapt if too hard for individual students or the class.

Hint: To help keep geobands from shooting around the room, have students cover the geoboards with their hands. Remove the bands keeping their hands over the boards until the bands are loose.

Moebius Strip

As you look for ways to engage your students in mathematics, explore the magic of the moebius strip. The moebius strip is a loop of paper with a half twist. Originally, the moebius strips model, with a twist, was used in car fan belts for more even wear. How many sides and edges does a moebius strip have? The twisted loop has several strange characteristics.

This site explains how to make a moebius strip:

mathforum.org/sum95/math_and/moebius/moebius.html

Have students predict and further explore a moebius strip with a full twist as well as using a wider strip and drawing a line one-third of the way in and then cutting on the line.

"Paul Bunyan and the Conveyor Belt," a story from the book Mathematical Magpies is a great opener to the moebius strip. A summary of the tale is available at this site:

www.andrews.edu/~calkins/math/webtexts/geom01ac.html

Directions are included at this site for the moebius heart with some other explorations that engage students:

www.questacon.edu.au/html/mobius_strip.html

Brazilian stamp

Escher used moebius strips in two of his designs. All nine ants are on the same side.

Escher ants

The moebius strip is used in the recycling symbol.

Recycling symbol

For other examples, see www.planetpals.com/recyclesymbols.html

Interested students may want to do further explorations in topology -- the study of properties of shapes that do not change when the shape is distorted:

www.questacon.edu.au/html/what_is_topology_.html

Art in Math

How can you incorporate art into the math classroom, or math into the art classroom? As the end of the school year approaches, such projects might be of interest to you and your students.

Here are some links to artists whose work has math connections. Marco Polo allows you to search names and concepts: www.marcopolo.worldcom.com

Piet Mondrian, Dutch painter (1872-1944)
www.ibiblio.org/wm/paint/auth/mondrian
www.artcyclopedia.com/artists/mondrian_piet.html

Wassily Kandinsky, Russian painter (1866-1944)
www.arts-studio.com/kandinsky
www.ibiblio.org/wm/paint/auth/kandinsky
www.artcyclopedia.com/artists/kandinsky_wassily.html

Victor Vasarely, Hungarian (1908-1944)
www.artcyclopedia.com/artists/vasarely_victor.html

Click on an image to enlarge and see the full set:
www.vasarely.org/virtual-gallery.html

Here is a lesson creating some op art based on Vasarely's work:
www.artcyclopedia.com/artists/kandinsky_wassily.html

Robert Delaunay, French artist (1885-1941) uses geometric shapes to create images:
www.robert-delaunay.net
www.artcyclopedia.com/artists/delaunay_robert.html

Yaacov Agam, Israeli sculptor, (1928-)
www.artcyclopedia.com/artists/agam_yaacov.html
www.artra.net
www.agam.net/museum/index.html

See the Moebius Random Sampler about Dutch graphic artist MC Escher and tessellations.

Palindromes

Many of you are familiar with palindromes. How can they be used in the math classroom? Here are a few math-related palindromes:

Here are some time and calendar examples:

Can you and/or your students think of others?

Here is a two dimensional palindrome:

G E L
E Y E
L E G

Can you create another one?

Does every number become a palindrome? Choose a 3 or 4 digit number. Reverse the digits and add to the original number. Take the new sum and reverse the digits and add. Keep reversing the digits and add until you reach a palindrome. How many steps does it take to get it to a palindrome? Can you find a 4-digit number that takes 5 or more steps?

For more information:

These websites have more information on palindromes and palindromic numbers:

This site also links to some palindrome books:

Palindromes help to create student interest and are useful in exercising problem solving skills in math classrooms.

Literature in Math

When literature is connected to math, students tend to solve problems in a more open-ended way. Math in textbooks sometimes lacks a connection or context for students. Here are some sources of engaging problems/lessons connecting literature and math.

The Ask Eric website has some interesting lesson plans based on children's books. Some of the lessons include the student worksheets.

Portland Public Schools' website describes some books with math connections which can be useful in developing math concepts.

Also, Van De Walle's book, Elementary and Middle School Mathematics, has recommendations for specific content strands.

Tessellations

Creating a tessellation with one or more shapes helps student explore geometric properties and develop problem solving skills. A tessellation is a tiling of congruent shapes without overlaps or gaps.

Encourage your students to design their own tessellations using Tom Snyder's Tessellation Exploration, (www.tomsynder.com), Tesselmania or Geometer's Sketchpad. Your students can submit their hand or computer generated designs to www.worldofescher.com/contest by April 1, 2002.

The works of MC Escher, Dutch graphic artist, incorporate tessellations and help students see connections between math and art. For more background and links to Escher's works, visit:

This site has links to some interesting visual illusions:

Olympic Games - Winter 2002

Middle level students are more involved in math when engaged. What better way to engage them than with the Olympic Winter Games? Students can pursue their own questions, events, and design their own problems. They can organize the data, make predictions, and verify what actually happens.

For starters, have students check out the official site for information and links:

www.saltlake2002.com

As the torch travels around the United States, students can follow along at:

seattletimes.nwsource.com/sports/olympics/saltlake_02/fansguide/torch.html

The official site also contains many facts about the torch relay.

This site has lesson plans as well as an interactive Torch Relay Map. Click on a state for links to information about the state.

www.uen.org/2002

In the curriculum math section is an ice hockey lesson as well as data on the 1998 Winter Olympic Games Medal Summary by country and gender.

This website has information about local athletes as well as information about Sydney Olympics in 2000:

seattletimes.nwsource.com/html/olympics

These sites also have information and connections:

As we go for the gold, please share your students' math problems with other NO LIMIT teachers and students. Let the games begin!

Computation & Basic Skills

As I come into your classrooms and buildings I hear many concerns about computation and basic skills. How do we develop mathematically powerful students?

Mike Schmoker in Results: The Key to Continuous School Improvement says, "We labor under the incorrect notion that students must master basic skills before they can learn higher-order skills or engage in complex activities. Studies in math, reading, and writing clearly demonstrate that the opposite is true:

"Students learn best when basic skills are taught in a vital, challenging context that makes the skills meaningful. The very thing that keeps students from achieving in these areas is the dry, irrelevant teaching strategies we often employ, especially with students who most need real challenges (Means, Chelemer, and Knapp 1991).

"Low-performing students stand to gain the most from approaches that incorporate basic skills into a complex, higher-order tasks and problems. Schools and effective programs have demonstrated that standardized test scores can improve significantly when challenging tasks and activities are used (Resnick, Bill, Lesgold, and Leer 1991; Schmoker and Wilson 993; Livingston, Castle, and Nations 1989; Pogrow 1988, 1990)." (pg. 71)

Students need to do sense making activities to develop number sense. Again, how do we develop mathematically powerful students?

Fractals

A fractal is a great example of a pattern. Fractals are everywhere, those bright, weird, beautiful shapes. But what are they, really? Fractals are geometric figures based on pattern, and have special properties.

There's lots of information on the Web about fractals, but most of it is either just pretty pictures or very high-level mathematics. Check out these fractals sites, to help them understand what the weird pictures are all about -- that it's math -- and that it's fun!

A Mandelbrot Fractal "Scientists have discovered that systems in transitional states between order and chaos possess certain patterns with unique, predictable qualities. These patterns are called 'fractals.' In essence, they are visual images or pictures of chaos at work.

"Common in art, science, and nature, they permeate all aspects of our lives. Scientists have discovered fractals in everything from the crystals the forming ice and snow to the orbits of the planets. Artists are using fractal patterns in their work, and computer buffs everywhere are wild about fractal, which can be created on most PCs."

From Fractals, The Patterns of Chaos by John Briggs, 1992.

Conundrums

Stories with logical holes or conundrums, can be a very effective classroom tool. (Conundrums are also known as lateral thinking exercises.)

Students try to figure out the scenario and what is wrong with this story by asking yes or no questions. They begin to think outside of the box and I get lots of practice with their names! Plus, this is a good way for students to work together and get to know each other quickly.

Here are some conundrums you can try with your students:

  1. Ian and Sylvia were found dead on the floor. The only clues are broken glass and water on the floor. What happened?
  2. There was a man who was afraid to go home because of a masked man. Who was the masked man?
  3. The woman arrived at the hotel and immediately knew that she was bankrupt. Why?

If you are looking for additional conundrums, check out "Show me the riddles" at www.riddlenut.com or "Brainfood" at rinkworks.com/brainfood/lattrick.shtml. Another resource for mindbenders and puzzles is Mind Benders: Adventures in Lateral Thinking by David Bodycombe.

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