Google authentication failed. Please try again later!
All Lesson Plans

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. Click here 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. Click here to learn how to make your own visual patterns using the Question Builder feature in Polypad.

polypad.org/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.

polypad.org/06dnKHzkt8lWaQ

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

polypad.org/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
polypad.org/e0IlBhYmhEQ8xw

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

polypad.org/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.

polypad.org/DSKNM94cHBEZ1Q

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

polypad.org/w5ATfQ3X7V47Q

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

polypad.org/K1N7qxz0cpGAQ

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

polypad.org/TKxo3MCHoLyYdA

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

polypad.org/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.

polypad.org/0OGW9IEfkUzA

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

polypad.org/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:

polypad.org/iYMZq3ekwRvfcA

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

polypad.org/8hDuZMPGsbtAYw

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

polypad.org/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.

polypad.org/3w3vEBPrjSe0mg

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

polypad.org/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 .

polypad.org/lrMQKFyKcMDRA

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

polypad.org/oxp1aXBXXDvkeA

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

polypad.org/9imfv6BOKQAdLA

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

polypad.org/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.

polypad.org/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.