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Visual Patterns

Overview

Visual patterns are a powerful tool to engage students in deep mathematical thought and exploration. The tools on Polypad can help make students' thinking visible to classmates and teachers.

Using Visual Patterns in Class

  • Share one of the Polypad canvasses below with students. Watch this video to learn how to share canvasses with students and for an overview of how students can share their completed work with teachers.
  • Invite students to create the next three pictures in the pattern. Copying tiles is important in these tasks. This video provides an overview of three ways to copy tiles in Polypad.
  • The auto-correcting answer boxes for figures 4, 5, and 6 provide immediate feedback to students about whether or not they continued the pattern correctly. Once they've done so, students can predict the number of objects in a larger figure number and then create an algebraic expression to represent the nth figure in the pattern.
  • The ability to easily change colors, add tiles, move tiles, and rotate tiles within Polypad can help students make their thinking visible to classmates. The video below shows one way in which the Polypad tools helps to make student thinking visible.
  • Some of the patterns below are 3D forms on a 2D space. Using the isometric background can help:

Visual Pattern Canvasses

Below are some visual patterns. Upon opening one in Polypad, select "Save Copy" in the Library panel to save a copy to your account. Watch this video to learn how to make your own visual patterns using the Question Builder feature in Polypad.

Pattern 1 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/NVcxQC7AlEJD8Q

The number of rhombuses increases by one in each step. One way to express the algebraic expression for the nth term of the sequence of triangles is 2n2n.

Pattern 2 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/06dnKHzkt8lWaQ

One way to express the algebraic expression for the nth term of this sequence is n2+2n+2n^2+2n+2.

Pattern 3 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/hwaOwjfuBN8Hw

This pattern show the Triangular Numbers. Each number of objects can be arranged into a triangle. One way to express the algebraic expression for the nth term of this sequence is [n(n+1)]/2[n(n+1)]/2.

The tenth triangular number, which is the sum of the numbers 1-10.
Triangular Numbers on Pascal Triangle
Pattern 4 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/e0IlBhYmhEQ8xw

One way to express the algebraic expression for the nth term of this pattern is 4n3.4n-3.

Pattern 5 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/JbZJokN2I5WSlg

One way to express the algebraic expression for the nth term of this sequence is n2+(n1)2n^2+(n-1)^2. Below are some different ways to view this pattern.

Pattern 6 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/DSKNM94cHBEZ1Q

This pattern is a different visual representation of the square numbers.

Pattern 7 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/w5ATfQ3X7V47Q

One way to express the algebraic expression for the nth term of this sequence is n2+4n^2+4.

Pattern 8 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/K1N7qxz0cpGAQ

One way to express the algebraic expression for the nth term of this sequence is 2n12n-1.

Pattern 9 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/TKxo3MCHoLyYdA

One way to express the algebraic expression for the nth term of this sequence is 2n12n-1.

Pattern 10 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/Z1inuQAibgLuA

One way to express the algebraic expression for the nth term of this sequence is 4n4.4n-4. The nth term can also be expressed as n2(n2)2n^2-(n-2)^2.

Pattern 11 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/0OGW9IEfkUzA

One way to express the algebraic expression for the nth term of this sequence is 4n.4n.

Pattern 12 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/xk0Pz9952175Q

This pattern shows the Tetrahedral Numbers. A tetrahedral number can be represented as a pyramid with a triangular base and three sides called a tetrahedron.

Each tetrahedral number is the sum of the triangular numbers. The image above shows 15 spheres on the bottom layer, 10 spheres in the lighter pink layer, 6 in the red layer, 3 in the orange layer, and finally 1 sphere on top. 1, 3, 6, 10, 15 are triangular numbers. 1 + 3 + 6 + 10 + 15 = 35, creating the 5th tetrahedral number.

One way to express the algebraic expression for the nth term of this sequence is (n(n+1)(n+2))/6(n(n+1)(n+2))/6:

Pattern 13 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/iYMZq3ekwRvfcA

The is the pattern of cube numbers and can can be expressed as n3n^3:

Pattern 14 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/8hDuZMPGsbtAYw

One way to express the algebraic expression for the nth term of this sequence is n3(n1)3n^3-(n-1)^3.

Pattern 15 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/F6YOES9gE2x24w

This pattern is sometimes called the pool or border pattern. The blue tiles represent the pool tiles whereas the red tiles are the border tiles. The number of border tiles for an "n by n" square can be expressed in many different ways.

Pattern 16 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/3w3vEBPrjSe0mg

One way to express the algebraic expression for the nth term of this sequence is 3n+1.3n+1.

Pattern 17 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/FJvCNIKUt2ejg

The ratio between the consecutive terms is constant and is equal to 3. One way to express the algebraic expression for the nth term of this sequence is 3n/33^n/3 .

Pattern 18 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/lrMQKFyKcMDRA

One way to express the algebraic expression for the nth term of this sequence is n(n+2)n(n+2).

Pattern 19 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/oxp1aXBXXDvkeA

One way to express the algebraic expression for the nth term of this sequence is 3n+1.3n+1.

Pattern 20 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/9imfv6BOKQAdLA

One way to express the algebraic expression for the nth term of this sequence is 4n+2.4n+2.

Pattern 21 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/ewc8KLnhFU5x1A

The light green triangles can be expressed as n2n^2. The sequence of the to dark green triangles can be expressed as 6n+96n+9. So, one way to express the total number of triangles for the nth term is n2+6n+9n^2+6n+9 or (n+3)2(n+3)^2.

Pattern 22 - Polypad - Mathigon
Virtual Manipulatives for Mathematics Classrooms: polygons, number and algebra tiles, fraction bars, tangram, pentominoes, and more.
https://mathigon.org/polypad/B6FnyqKQdhSZhg

One way to express the algebraic expression for the nth term of this sequence is 2n2+n2n^2+n.

Credits

Some of the patterns on this page come from visualpatterns.org, created by Fawn Nguyen, and licensed under a Creative Commons Attribution 4.0 International License.

Have a visual pattern you've created you'd like added to this list? Email the Polypad link to teachers@mathigon.org.