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Organizing Patterns in Mathematics

Finding and understanding patterns is crucial to mathematical thinking and problem solving, and it is easier for students to understand patterns if they know how to organize their information.



Struggling students, including those with disabilities, can find it helpful to organize the information in a problem. This process requires them to think more deeply about each piece of information, and to think about how those pieces fit together.

Creating an organized list, table, chart, or other graphic organizer is an important problem-solving strategy (seeUDL Checkpoint 6.3: Facilitate managing information and resources(opens in a new window)). Organizing information will help your students to see patterns and learn mathematical concepts more easily. It will also help them to think about what they do and do not know about a problem. The strategies below provide a strong starting point for differentiating instruction.

Using Evidence-Based Practices

Teachers can help their students organize information to find and use patterns by using three categories of evidence-based practices: provide clear explanations(opens in a new window), give students strategies and models(opens in a new window), and provide ongoing formative assessment(opens in a new window). The chart below provides a concrete teaching strategy for each category.

Evidence-Based Practice Strategy

Introduce this strategy by presenting a list, table, chart, or other graphic organizer (using a spreadsheet or concept mapping tool) and point out why each kind of organizer is helpful. For example, it might help you see patterns, quickly find a particular example or piece of information, and/or make sure you are not missing any examples.

Explain to your students that organized lists are fairly easy to create as long as there are not a lot of items. Show them an example of the kind of list that can be created from a problem, and explain how the list can be used to organize the information in the problem.

Draw on a two-part strategy to support students who are not being systematic in how they organize information. First, explore whether the organizer they are using is appropriate. Next, ask them questions that will help them find a more systematic way of organizing information. For example, you could ask them the following questions: What are you trying to find out? How will this organizer help you? What might be a problem? Explain to me the system you’re using to make your list, chart, map, and so on. How do you know what to do next?

Technology Resources

You and your students can take advantage of both online and offline tools to create, analyze, and understand patterns, as this chart shows. 

A range of technology tools can greatly increase the organizing options available to students. Computer-based tools—such as spreadsheets and concept mapping tools (e.g., in a new window))—can check that data is organized correctly, make information easier to sort, and save time, especially when creating charts.

One effective approach for helping students organize information is to use concrete materials, often referred to as manipulatives. These tools help students and teachers represent abstract mathematical concepts. They also help students to link these concepts to prior knowledge.

Teachers can also move beyond physical manipulatives and provide students with technology tools. Virtual manipulatives are digital “objects” that resemble physical objects. Examples of virtual manipulatives include Base-10 Blocks(opens in a new window), Cuisenaire Rods(opens in a new window), and tangrams. Most virtual manipulatives come with structured activities or suggestions to help teachers use them in the classroom.

In the Classroom

Mr. Rutter’s Grade 6 class is working on a multiplication unit that focuses on multiples and factors. He is planning an upcoming lesson that will focus on finding greatest common factors. Understanding these concepts is important for understanding fractions, which is their next mathematics unit.

To ensure his students are college and career ready, Mr. Rutter’s teaching is guided by the Common Core State Standards Mathematics Practice Standard 7: Look for and make use of structure. Within this wider context, his more specific lesson objective is to have students learn organizing tools for working with factors and multiples.

Mr. Rutter plans to use a variety of technology tools in this lesson:

  • An interactive whiteboard to communicate visually with the class
  • A calculator
  • The Factorize applet(opens in a new window) to use as a computer-based alternative for organizing factors
  • The Factor Tree virtual manipulative to use as an extension activity and reinforce knowledge of factors

In order to assess his students, Mr. Rutter will question them about the completeness of the list of factors, observe the answers displayed on individual whiteboards, and guide student summaries about organizing and finding common factors.

His lesson plan is divided into three sections—launch, learning task, and closure—and is outlined below.

Lesson Plan


  • Introduce the lesson by reviewing the relevant terms.
  • Have students do a simple practice problem and write their answers on individual whiteboards.
  • Work on another practice problem as a class.
  • Explain to students the importance of organizing their work.

Learning Task

  • Demonstrate one method for organizing factors (factor rainbows), e.g.,

  • Walk students through the process of completing a factor rainbow.
  • Once the factor rainbow is complete, discuss with the class how to know if it is complete.
  • Demonstrate a second method for organizing factors, using an online tool.
  • Have students choose a strategy and practice finding all the factors of a couple of numbers.
  • Review how to find common factors, including the greatest common factor.


  • Have students write a summary about what they learned in the lesson.
  • Have students copy the summaries into their notebooks.
  • Begin work on homework problems.
  • Have students who finish early explore another virtual manipulative a

Online Teacher Resources

This article draws from the PowerUp WHAT WORKS(opens in a new window) website, particularly the Organizing Instructional Strategy Guide(opens in a new window). PowerUp is a free, teacher-friendly website that requires no log in or registration. The Instructional Strategy Guide on organizing includes a brief overview with an accompanying slide show; a list of the relevant mathematics Common Core State Standards; evidence-based teaching strategies to differentiate instruction using technology; short videos; and links to resources that will help you use technology to support mathematics instruction. If you want to dig deeper into the research foundation behind best practices in the use of virtual manipulatives, take a look at our Tech Researc h Brief(opens in a new window) on the topic. If you are responsible for professional development, check out PowerUp Your Professional Development (opens in a new window)for helpful ideas and materials for using the organizing resources. Want more information? See PowerUp WHAT WORKS(opens in a new window).

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