Valentine Problem Solving: Heart Paths

Heart Paths

If students enjoyed the Pascal snow activities, then they'll also enjoy this Valentine's version of Pascal paths.  Heart Paths challenges students to find all the different paths that spell HEART if they can only move from a letter to one of the two letters directly below it. 

The number of possible paths is linked to Pascal's Triangle and older students will be amazed at how this mathematical pattern can be used to help them identify all of the paths.  A recording sheet is included in the packet so that students may easily record paths and identify duplicates.

Download the Heart Paths activity which includes the problem, directions, recording sheet and answer key.


Suggestions for Classroom Use:

• Make an overhead of the Heart Paths problem or use the PDF on a SmartBoard for presentation.  
• Demonstrate a legitimate path moving from a letter to one of the two letters directly below it.
• Demonstrate an illegitimate path by moving from a letter to another letter not directly below it.
• Do not tell students how many paths there are.  Students need to develop confidence in their own ability to capture all of the paths.
• Use an overhead of the Heart Paths recording sheet (or the PDF file on a SmartBoard for class discussion.  Have students come to the overhead to draw possible paths.
• Follow up with a discussion of systematic counting.  If students did not use a systemized approach to this problem, demonstrate finding all of the possible paths for the first E, then move to the second E, etc.  NOTE:  The answer key included in this activity follows a systematic counting system.  Systematic counting is an important concept in discrete math and very applicable to computer programming.  For students, it is a logical approach that almost guarantees that they will consider every option, so it is a strong strategy to add to their problem solving repertoire.

Valentine Quilt

This Valentine Quilt is a great culminating activity for a study of the geometry of quilts. Each student designs a 9-square paper quilt patch using only red, pink and white paper. Students may elect to use a traditional quilt pattern or design their own original pattern and name it. Assemble all of the quilt squares into a class quilt. 

Be sure to use the Quilt Block Challenge to introduce students to the mathematical geometry of quilts.  Here are some additional Mathwire handouts to help you get started on any quilting study:

• 3x3 Quilt Patterns  may be used as templates for copying shapes onto colored paper and for students to assemble their own quilt blocks.
• Paper Quilt Block Writing to Learn helps students reflect on the symmetry and transformations found in their quilt block design.

These internet sites are also helpful when introducing students to the mathematics of quilting:

• Annenberg's Quilts addresses 4 different kinds of symmetry found in quilt blocks. Students will then have to identify the kind of symmetry used in other quilt blocks, then select pieces from the four given pieces to replicate that quilt block. 
• Nine Patch Quilt Block Patterns  shows students pictures of traditional quilt blocks based on the 3x3 grid.

Math-Literature Connection: Quilting

A study of quilts offers the chance to investigate tessellating shapes and an opportunity to apply transformational geometry as students slide (translate), flip (reflect) and turn (rotate) quilt pieces to create a traditional quilt or to create a tessellating quilt.   Beyond the pure geometry, the use of color suggests different shapes within a shape and contributes to the beauty of the quilt design. 

Quilting books offer a good interdisciplinary introduction to the mathematics of quilting.  Consider using some of these books, as appropriate, to introduce students to the richness of quilting patterns and color.  Subsequent mathematical activities will introduce students to the geometric shapes and tessellations inherent in quilt blocks.

Quilting Books

Quilting is a folk craft that crosses all times and cultures. Some books discuss the traditional American patchwork quilt and its sentimental value to families. Quilts offer a great opportunity to include multicultural topics to discuss mathematics and the contribution different cultures have made to this art form. 

If it is true that mathematics is the study of patterns, then quilting offers a rich tapestry for this study. Students of all ages learn to appreciate the mathematical patterns found in traditional quilting designs. These books enhance the mathematics with heartwarming stories of how quilts truly are the fabric of our lives. 

See Mathwire's list of books about quilting.   Most of these books are readily available in school and/or neighborhood libraries.  Select books that coordinate with literature or social studies units for an interdisciplinary approach.

Quilt Square Challenge

This activity is designed to help students develop spatial sense by decomposing shapes into smaller units. Students are shown a quilt square and must use their small quilt pieces to create that design in either the 4-square or 9-square mat by sliding and turning the quilt pieces to achieve the desired image.   Students should assemble the quilt pieces on either the 4-square or 9-square mat which helps them organize their work. 

The Quilt Square Challenge was originally designed for a middle school unit on transformations. Since then, students as young as first grade have enjoyed the challenge and mastered the art of transforming quilt pieces to produce the desired design. It is amazing to watch students as they maneuver the pieces and improve their spatial sense over the course of the activity.  

 Classroom Management:

Make overheads of the quilt patterns or use the pdf file with an LCD projector so that students see a large visual of the pattern. Provoke discussion with students about how they see the pattern. Some students reproduce using the black spaces; others see the white spaces. Many students see a pattern (e.g. tree, boat) that helps them reconstruct the pattern. Other students scan square by square and reconstruct the design one square at a time.  

Provide adequate time for students to complete the different patterns. Some teachers check students' completed work, then allow these students to "mess around" with other designs while other students are finishing the challenge design.

Read more about Quilt Square Challenge including additional classroom management suggestions, lesson plan, and materials.  The Mathwire page includes links to download all templates designed by Terry Kawas for Mathwire.com.

Problem Solving: Remainders in Division

Snow Day Signboard and School Closed Signboard encourage students to look for patterns in repeating letters to figure out which letter will be the 100th letter to be repeated on the signboard.   Students may use division and remainders, skip counting or repeated addition to solve the problems, making them accessible to students in many grades.

Each problem includes a challenge to extend the problem-solving experience and a possible solution. 

• Download Snow Day Signboard.
• Download School Closed Signboard.

Pascal Paths

How Many Ways Can You Make Snow? encourages students to apply the patterns in Pascal's Triangle.   A teacher instructional plan with mathematical background, answer and challenge is included to explain how to present this problem.   A recording sheet was also included as part of the teacher packet so that students are able to record all different solutions in a "systematic" way, which is a goal of discrete mathematics. 

• Download How many different ways can you make the word SNOW? problem, recording sheet and answer key. 

Extension:
Extend this lesson using the word winter to form Pascal paths.  How many possible different paths are there for this word?  How does it relate to Pascal's Triangle?  

How Many Winter Paths  involves systematic counting of the different Pascal paths in this arrangement of the word WINTER.   Students will be challenged to relate this activity to Pascal's triangle as they analyze their solutions to see if they have found all of the ways to reach each R in the bottom row.   A teacher instructional plan with mathematical background, answer and this challenge is included to explain how to present the problem.   The recording sheet encourages students to trace one path in each frame, making it easy to see if students use a systematic counting approach to solving the problem.

• Download How many winter paths do you see? problem, recording sheet and Pascal's link.  This PDF also contains instructional suggestions for presenting this problem to students.

 Note:  Systematic counting is an important concept in discrete mathematics.  Be sure to model using a systematic approach to finding all of the different paths, as presented in the instructional suggestions for this problem.  Students will benefit from adding this logical and systematic counting approach to their problem-solving repertoire.
 

Penguin Bowling Game

The Penguin Bowling Game was designed to provide mixed basic facts practice. Students toss 4 dice, then use the numbers from the dice throw and any of the four operations (+, -. *, /) to form expressions to knock down penguin pins. They are trying to create expressions that result in answers of each number from 1-10.  Each student uses a recording sheet to write their expressions.  Insert the recording sheet into a plastic sheet protector and use dry erase markers for a renewable option.

Students use a score sheet to record group scores as one does in real bowling games. Remember to model the scoring process, using an overhead of the scoring sheet, if students are not familiar with scoring regular bowling games. Teachers will especially have to model how to score a strike, as students will be strongly motivated to knock down all ten penguins in each round to gain these extra strike points.

This activity is a great do-now for students at the beginning or end of class to provide basic facts practice that students find highly motivating in a game format.  Additionally, students profit from seeing and hearing the expressions their peers created and they are more likely to incorporate similar strategies in future play.

Penguin Bowling Game Materials:

• Download the recording sheet and directions.
• Download the score sheet.

Winter Fraction Words

This is a fun way to practice fractions with a little winter play on words.  Students read each clue and enter the letters in the appropriate boxes.  If they correctly calculate the fractional parts, the box should spell a wintry word.

Extension:  Challenge students to create their own fraction word collections.  After completing their collection, they should ask classmates to edit their creations and verify the correct clues.  Add these student-generated activities to the classroom math center or use them as a do-now for quick fraction review.

Download Winter Fraction Words.

Snowflake Symmetry

Symmetric Snowflake
 
Here's an activity for all those students enjoying snowy weather.  Symmetric Snowflake  challenges students to use pattern blocks to complete the other half of a snowflake, based on the pattern to the left of the given line of symmetry.  Afterwards, students are asked to create their own symmetric snowflake, using pattern blocks.   

• Download Symmetric Snowflake.

Snowflake Symmetry

Challenge students to predict and draw what a snowflake will look like when the given pattern is cut out. Then, have students actually cut out the snowflake pattern to check their prediction. Finally, students will identify and draw in lines of symmetry for the snowflake pattern. Train students to think about and visualize the symmetry inherent in this hexagonal structure. 

• Download Snowflake Symmetry.
• Enrichment:  Provide copies of  Snowflake Symmetry Student handout so that students can record their cut design and challenge their classmates to predict what the finished product will look like. 

• Bulletin Board:  display both the folded designs and the finished products in random order. Label each design with a number and each finished product with a letter. Challenge students to correctly match each pair.

The M&M Game

M and M Game: 2-dice Version 

Students place M&M markers on the numbers 2-12.   Students may place one M&M marker on each number or place several on some numbers and leave other numbers blank.   Next, students toss two 6-sided dice, find the sum, and remove an M&M marker from that number, if there is still one.   The first player to remove all markers wins the game. 

This game helps students develop an understanding of the probability of a two-dice throw.  As students play the game, they practice addition facts, but they also learn that some sums are tossed more frequently than others.  This hands-on experience is then solidified with a discussion of all of the possible outcomes, creating a chart to visually capture the data.

Next, students play the game again with this mathematical knowledge and discuss how this information impacts the placement of M&Ms.  Couple this game-playing with class data collection and students may then compare theoretical and experimental probability of a larger sample.

This is just one example of how a simple game may be used as a data collection activity to develop student understanding of theoretical and experimental probability concepts.  Some teachers question the use of games in the classroom pressed for time.  In my experience, this rich use of games to practice basic facts AND as a data collection activity is time well-spent and students respond positively to both the practice and the discussion of these hands-on, fun experiences.

NOTE:  This game, created by Susie Siegel as a grad school project, uses a sentence strip as the number line.  The M&M pieces were created using PowerPoint.  Teachers may choose to laminate the pieces for use in a classroom math center.

MATERIALS:

• Download Ms. Siegel's M&M Game directions and recording sheet.
• Download a picture, alternate recording sheet and directions for using that recording sheet.  NOTE:  This recording sheet was designed to fit into a sheet protector so that students may use dry erase markers to play and record data.  This option may be used without the M&M game pieces.
• Download the 2-dice toss recording sheet for a discussion of the theoretical probability of tossing two dice.  Students write one outcome that produces that sum in each box to create a visual display.   NOTE:  use different colored dice for this activity so that students are able to see that tossing 4 on the red die and 3 on the blue die is different from tossing 3 on the red die and 4 on the blue die.  Both must be counted as outcomes.  Students might benefit from actually using two different color markers to record these outcomes as well.
• Download M&M pieces for use in playing the game.

Penguin Math Games

Capture the Penguins Game uses the outcome of two-dice toss to form a coordinate pair. Students toss two dice (one regular and one A-F) in this fun game that introduces students to coordinate graphing in the spaces.   Students form a coordinate pair based on the dice toss and capture a penguin, if possible.   Students use the accompanying recording sheet to keep score during the game. 

• A-F Dice: Create A-F dice using plain dice or purchase small wooden cubes at a craft store to make the dice.  
• Penguin Pieces: Penguins pictured to the right were created by painting wooden clothespins and doll stands, both readily available at craft stores. 

Download Capture the Penguins game mat, instructions and student recording sheet. 

Free the Penguins Game:  The penguins are stuck on the ice floes. Only a roll of the die can free them to search for food in the ocean. Students will practice addition or subtraction facts as they try to be the first to free their penguins. Use mats with clothespin penguins for the most fun! 

These games are not only an excellent opportunity to practice basic facts but they allow students to collect data and reflect on the probability inherent in the game. Suggestions are included with this game for students to keep tallies of all dice tosses, organize the class data in a line plot, and then analyze the data for patterns and trends. 

Download Free the Penguins game mat, instructions and suggestions for data collection.

 

Penguin Math-Literature Connection

The family in this book receives a penguin in the mail each day and these penguins quickly add up to take over the house and their lives. The book already contains measurement calculations, but the book also lends itself to some problem solving possibilities. After enjoying the book, continue with some of these problem solving activities, created by Mathwire:

• Students need to apply calendar skills to solve the Penguin Delivery problem.
• Dad dreams up another crazy arrangement for the penguins in Penguin Formation .
• See more penguin problems including Penguin Parade and Penguin Puzzler which also feature patterns that would have intrigued Dad.

Pascal's Penguins

Pascal's Penguins is an effective introductory activity to the well-known mathematical pattern known as Pascal's Triangle. Students must look for patterns in these penguin variations of Pascal's Triangle. The activity challenges students to identify patterns, fill in the missing numbers and write the next line in the pattern. Class discussion should encourage students to share all of the patterns they see in Pascal's Triangle and discuss how these patterns helped them discover the missing numbers. 

Pascal's Penguins - 1 introduces primary students to a small version of Pascal's Triangle in this simple patterning activity.  

Pascal's Penguins - 2 introduces a larger version of Pascal's triangle and encourages students to identify the different patterns within the triangle and use these patterns to fill in the missing penguin numbers.