Heart 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.  Students use the Heart Path worksheet to record each path, making it easy to check for duplicates.


This problem is best solved using systematic counting, an organized approach to solving the problem.  While many students might not use this approach independently, it is beneficial for the teacher to spotlight students who successfully used this approach.  Or, the teacher might model the approach as students check their own solutions, adding paths they might have missed in a more haphazard approach.


Download the Heart Paths worksheet which includes the problem sheet, worksheet and solution.

Candy Corn Pattern

Candy Corn presents a triangular numbers problem using a candy corn pattern.  Younger students might use candy corn or other manipulatives to model the problem.  A sample solution shows how older students might use an input-output table to model the pattern and find the solution without the use of manipulatives.

• Download the Candy Corn pattern problem.

Pascal's Pumpkins

Pascal's Pumpkins   encourages students to look for patterns in Pascal's Triangle.   The handout develops awareness of this important mathematical pattern through a timely seasonal activity.

Students who take higher math courses will meet Pascal again in many different applications, including probability.   This handout is an outgrowth of the Rutgers Discrete Math Institute.

Download Pascal's Pumpkins handout and solution.

Problem Solving: Number Patterns

Mathematics is full of patterns and mathematicians analyze data, searching for patterns that will allow them to draw conclusions and make hypotheses.

These problems were specifically designed to reinforce common mathematical patterns.  Some patterns are easily identified.  Others are best solved using an input/output table to collect and analyze mathematical patterns.

Grades 1-2:

• Bay Street requires students to analyze the pattern of house numbers on Bay Street. Students write down the pattern they see and use this number pattern to write in the missing house numbers. 
• King Street requires students to analyze the pattern of house numbers on King Street. Students write down the pattern they see and use this number pattern to write in the missing house numbers.  

Grades 3-4:

• Baseball Season is a pattern problem that can be easily solved using a table of values. 
• Anthony's Allowance provides additional pattern practice.  

It is important that students identify the patterns they see, using their own words to describe these patterns.  Supply mathematical terms as you summarize their findings, adding terms such as odd, even, Fibonacci sequence, repeating pattern, growing pattern, etc. Add these terms to the math word wall so that students may consult this list in future problem solving.

Hundred Board Puzzles

The hundred board or chart holds a prominent place in primary math classrooms.  The chart is a visual representation of the patterns in our mathematical system of numbers.

Students are able to learn the patterns in the hundred board by assembling puzzles.   Teachers are able to assess student use of patterns in rows and columns by observing the student at work.  

Differentiation:

 This task is easily differentiated to accommodate the varied levels in a first grade class by changing the number of pieces and the shape of the pieces.   Puzzle bags should be sequentially lettered so that students progress through harder versions of the task.  

Extensions:

• Have students and their families create their own hundred board puzzles.
• Challenge students to fill in missing numbers on hundred board pieces or whole hundred boards as practice using the patterns in the hundred board.

Materials:

•  Download a hundred board template.

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.
 

Investigating Growing Patterns

Introduce elementary students to the concept of functions by investigating growing patterns. Visual patterns formed with manipulatives are especially effective for elementary students and allow them to concretely build understanding as they first reproduce, then extend the pattern to the next couple of stages. Finally, students explain the pattern in words and try to write a rule that works for any stage.

See Mathwire's Investigating Growing Patterns for specific suggestions on using these patterns to help elementary students develop a concept of functions. The article includes activities with handouts that teachers can use to introduce this topic to elementary students. A Function Scrapbook of function ideas is included and examples of function problems created by elementary students will be added as they are completed. Literature connections and web links to relevant sites on the internet are also included in the article.

Pattern: Triangular Numbers

Bat Jamboree by Kathi Appelt introduces the triangular number pattern as bats assemble for the final number beginning with 10 bats in the bottom row, 9 in the next row, etc. to the very top row with 1 bat. Students are introduced to the 55 bats in formation and their various acts but the book "isn't over until the bat lady sings."

Students will enjoy this introduction to an important mathematical pattern. Teachers can find many problems that build upon this triangular number pattern and extend the experience.

• Annual Fall Parade challenges students to use the triangular pattern to figure out how many students are in the fourth grade. Given the number of full rows, students must apply the pattern and use effective recording (picture, table, etc.) to explain their reasoning.
• Candy Corn presents a triangular numbers problem using a candy corn pattern. Younger students might use candy corn to model the problem. A sample solution shows how older students might use an input-output table to model the pattern and find the solution without the use of manipulatives.
• See more Bat Activities in Mathwire's Math Activity Themes:  Bats collection.

12 Days of Christmas

Each year PNC Bank calculates the cost of the gifts in this familiar Christmas carol.  Watch a short video that presents each gift and tells the percent increase or decrease for that gift each year.  Following this presentation, users may select from a number of games available on the site or view a graph of the annual cost index.

Investigate the math behind this holiday gifting by working through DIMACS The Twelve Days of Christmas and Pascal's Triangle.  Students who struggled to figure out the total number of gifts received will be astounded to discover that the patterns in Pascal's triangle may be used for an easy solution.

Students might also enjoy singing 12 Days of Math, a math teacher's parody of the famous carol. 

Print out pictures of the 12 Days Gifts for students to color and/or use as props when singing the song and discussing solutions to the problem of total gifts given in this holiday song.

Bat Jamboree

Bat Jamboree by Kathi Appelt introduces the triangular number pattern as bats assemble for the final number beginning with 10 bats in the bottom row, 9 in the next row, etc. to the very top row with 1 bat. Students are introduced to the 55 bats in formation and their various acts but the book "isn't over until the bat lady sings." Students will enjoy this introduction to an important mathematical pattern. Teachers can find many problems that build upon this triangular number pattern and extend the experience.


Student Written Problems: ask students to write original problems that use the triangular number pattern. Being able to write similar problems and solve them require higher-order thinking skills as students apply, synthesize and evaluate both the problems and the solutions.


Check out more Bat Activities in Mathwire's Math Activities Themes collection.

Pascal's Pumpkins

Pascal's Pumpkins encourages students to look for patterns in Pascal's Triangle. The handout develops awareness of this important mathematical pattern through this timely seasonal activity. Students who take higher math courses will meet Pascal again in many different applications, including probability. This handout is an outgrowth of the Rutgers Universiy Discrete Math Institute.

Download Pascal's Pumpkins.  The PDF file contains the student handout and a detailed explanation of the different patterns students might find.  Be sure to make an overhead copy of the pattern so that students are easily able to point out patterns and explain the pattern that they see as they fill in the empty pumpkins.

Extend the activity by asking students to draw and label the numbers in the 7th, 8th, etc. row.  How does identifying the patterns help them complete this task more easily?