Monday / December 4

Helping Students Connect Manipulatives to Paper-Pencil Math

When we’re being honest about using math manipulatives, the number one concern I hear revolves around management – students simply don’t use them as intended. They urn these tools into something playful or even dangerous. You can learn what to do about that in my previous blog post, Managing Manipulatives During Math Class…From Playful to Purposeful.

The number two concern I hear centers on connections. Students use manipulatives to explore a concept one day, and then, the next day, they don’t transfer their understanding to paper-pencil work. After putting so much effort into the planning and preparation for hands-on lessons, this can be both disheartening and frustrating. That said, you can take control of this situation with a simple strategy called Side-by-Side Math.

Side-by-Side Math helps students connect the understanding they gained from using manipulatives to the work they produce with paper and pencil. As they work in pairs, students coordinate their thinking by discussing what they are doing as they solve the problem in different ways at the same time.

Here’s how Side-by-Side Math works:

  1. Put students in pairs. Although it’s tempting to put them in groups-of-three so that they can each choose one type representation (physical, visual, symbolic), this process works much better when two students sit side-by-side, coordinating both their actions and their descriptions.
  2. Provide manipulatives and white board for students to use. I prefer to let students select the manipulatives they want to use rather than only providing only one option. I also typically encourage multiple representations on the white boards, whether visual (e.g., number lines, number bonds, tables, etc.) or symbolic.
  3. Give the students a task to work on together. This can be as simple as an arithmetic problem or equation to solve. Be sure to have extras on hand so pairs who work quickly will always have more to do.
  4. Ask students to sit side-by-side. One student works with the manipulatives and the other records on a white board, using either visual or symbolic representations. Sitting side-by-side allows both students to see their work right-side-up.
  5. Listen as students coordinate their actions and descriptions. They talk, record, observe and internalize how the different representations correspond to each other.

Here are examples from each of three different grade levels and concepts, using KP Ten-Frame Tiles. You could also use counters on ten-frame mats.

Example 1: Make-a-Ten

6 + 8 with KP Ten-Frame Tiles

6 + 8 with symbols


6 + 8 with KP Ten-Frame Tiles

6 + 8 with symbols


6 + 8 with KP Ten-Frame Tiles

6 + 8 with a number line

Example 2: Commutative Property of Multiplication

9 x 3 with KP Ten Frame Tiles

3 x 9 with KP Ten Frame Tiles

9 x 3 and 3 x 9 with sketching


9 x 3 with KP Ten Frame Tiles

3 x 9 with KP Ten Frame Tiles

9 x 3 and 3 x 9 with number lines

Example 3: Solving Equations 

subtract x from each side

subtract 2 from each side

solution with symbols


When you do this for the first time, you’ll want to be explicit about what the process looks like and sounds like and you can create an anchor chart to help students remember the routine.

Sample student anchor chart to describe the explicit steps

Whether working with young learners or high-school students, they typically benefit from practice as they prepare for using the Side-by-Side Math strategy. This is a great way to emphasize your speaking and listening standards.

I encourage you to try this out. You’ll be amazed at how students make connections across multiple representations when you provide them explicit opportunities to witness and discuss the relationships.

Written by

Kimberly Rimbey serves as the chief Learning officer at KP Mathematics. A lifelong teacher and learner her heart’s work centers on equipping teachers and helping them fall in love with teaching and learning over and over again. Kim’s interests include high-quality professional learning modules, building conceptual understanding through multiple representations and meaningful discourse, and building pedagogical content knowledge that goes beyond the theoretical and into the classroom. Always a teacher at heart, Kim has served in many teaching and leadership capacities within school districts, organizations, and the private sector. She is the co-inventor of KP® Ten-Frame Tiles and has authored and co-authored such publications as Mastering Math Manipulatives and Meaningful Small Groups in Math for Corwin.

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