Developing Algebraic Thinking Through Visual Patterns

Visual Patterns is a very simple and wonderful website, created by a public middle school teacher in Southern California named Fawn Nguyen.

The site is essentially a collection of 157 different visual patterns (and growing). For each pattern, you are given the first three figures/stages of the pattern. For example, here’s #25:

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You are also told something about the 43rd figure in the pattern. For example, the 43rd figure in the pattern above has 89 squares.

That’s it, but that’s all you need.

When you ask adult education students what algebra is, many will say things like “x and y”, “inverse operations”, “variables”, “negative numbers”, or “solving for x”. They will also say things like “heartbreak”, “when I left school”, “something that has nothing to do with real-life”. Their answers to the question often can lend insight into the root of their struggle. Algebra is a way of thinking, that involves core concepts like equality/balance, generalizations and symbolic representation, but students are often taught algebra as only a set of steps to perform. That is  similar to being taught shorthand before you learn to write.

There are lots of things you can  do with visual patterns like the 157 you’ll find here, especially in the development of algebraic thinking. By exposing students to visual patterns regularly, and guiding their explorations, you can help them see that algebra is more than solving for x and that it is not just a series of procedures you have to memorize. You can use students’ own observations to build a concrete understanding of concepts like constants, variables, solving for unknowns, equivalence between equations and equation writing. Students at all levels of adult education can and should be looking at patterns and making observations and eventually predictions and generalizations.

Here’s an example of a line of questioning you might use with the pattern above (which can also be used with almost any other visual pattern):

  1. Sketch the next two figures.

  2. How are the figures changing?

  3. Create a chart with the data you have gathered. (This can help students identify the iterative rule of the pattern. It also gets students to organize the data they have gathered, which can help them find patterns. It also allows them to create a representation of the pattern in the pictures.)

  4. In a few sentences, describe what the 10th figure would look like.

  5. Explain how you would figure out the number of squares in the 99th figure. (This kind of question encourages students to look for an explicit rule. For example, consider the pattern #25 above. You can use the fact that each figure has two more squares than the one before it to think about the 10th figure, but that becomes less efficient for the 99th. Of course, teachers can use a smaller number for lower level students or when introducing visual patterns.)

6. In a few sentences, describe how to figure out how many squares there would be in any figure in this pattern. (Teachers can use the question as a leaping off point for more formal equation writing. Also, one can develop different equations for most of these patterns, which can be a leaping off point for looking at equivalent equations.)

  1. Teachers can also switch the question around. So for instance, for the pattern above you can ask, “Which figure/stage will have 49 squares?” (This type of question can be the foundation of solving for an unknown, except instead of us having to tell students how to do it, they can tell us.)

  2. Teachers can also extend the problem by asking a question like, “What will the perimeter of the 52nd figure be?”

I have had a lot of success in adult ed classrooms asking questions like the ones above.

Nguyen offers a few different versions of handouts with student questions on the Visual Patterns Teacher page. Those handouts have fewer, but similar questions, but they ask students to write equations. Many adult ed students won’t be able to do this on their own, at least in the beginning. The line of questioning above is designed to help them make the connection between the concrete picture and the abstract equation.

Here are a few examples of other patterns from the collection:

#44 Visual Pattern #44

#85 Visual Pattern #85#89 is a link that actually takes you to the fourth stage of a pattern of interlocking rings, where you can rotate a three dimensional model.

#119 Visual Patterns #119

#154 Visual Pattern #154

Once students have some experience working with visual patterns and start to get the hang of it, they love the opportunity to create their own. If you look under the “Gallery” tab at Visual Patterns, you will see a selection of student-created patterns. Here’s an example of one I particularly enjoyed:

student visual pattern

The answer key (by which I assume she means the equations) is not available on the site, but if you contact Fawn, apparently she will send it to you.

Here are just a few ways teachers might use this site with students:

  • You can use these as warm-ups for the beginning of class.
  • You can use them as a source of problems in a unit on patterns and algebraic thinking
  • One students are comfortable with the explorations, you can even send them to the site and have them work – either on patterns you assign, patterns of their own choosing, or both.

For more ideas about how to make the connection between patterns and algebraic reasoning, I recommend reading “Developing Algebraic Thinking Through Pattern Exploration”[1] by Leslee Lee & Viktor Freiman. It models an exploration of a particular pattern and also offers a general line of questioning (similar to the one above) that can be used with a wide variety of patterns.

 

 

[1] From Mathematics Teaching in the Middle School, Vol. 11, No. 9, May 2006.

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About Mark Trushkowsky

Mark enjoys doing math problems that take weeks, family sing-a-longs and reading late into the night. At 16, he believed the next revolution would be waged through poetry. Now he believes it is adult basic education. But he still likes poetry. Mark has worked in adult literacy and HSE since 2001. He is a founding member of the NYC Community of Adult Math Instructors (CAMI). He was born and raised in Brooklyn where he lives happily ever after with his partner Sarah and their daughter Liv. Follow me on Twitter (@mtrushkowsky)

8 thoughts on “Developing Algebraic Thinking Through Visual Patterns

  1. I am planning for a new (to me) math only class and I can’t wait to explore these materials, ideas and manipulatives on something bigger than my cell phone.

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  2. Fawn also has a great blog (http://fawnnguyen.com/) where she tells stories about teaching and learning mathematics in the classroom. For more ideas about how to use visual patterns to develop your students’ algebraic thinking and their understanding of functions, check out her post called Grade 6 Rocks Visual Patterns – http://fawnnguyen.com/grade-6-rocks-visual-patterns/. The post has some great ideas from other teachers about how they have been using VisualPatterns.org, as well as some cool examples from Fawn’s own class.

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    1. Thanks for posting this, Mark! It’s an exceptional one-stop-shopping resource that contains numerous visual models for thinking about patterns at multiple levels.This website has given me lots of ideas, and I definitely see this as a weekly go-to resource. For example, I’d like to start my students out with something like Pattern #14, which illustrates a linear relationship and then eventually bridge to Pattern #141 (the quadratic relationship we explored at the Math Institute). .

      I also appreciate the model of questions that you’ve included in your review, because if used consistently with each new visual model, students will have that “something familiar” to ground them with each new (and maybe more complicated) pattern.

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  3. I wanted to share a blog post by Nat Banting, who has done interesting things with Vine (animation/short movie app) and visual patterns. Nat talks about how he started presenting visual patterns in short videos because of confusion his students had seeing three stages of a figure on paper (a static medium). The video brings home the point that visual patterns (and functions) are about change.

    http://musingmathematically.blogspot.com/2013/08/animating-patterns.html

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