Visualizing the Difference Between Addition and Multiplication

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

Having concrete and representational experiences are essential if students are to build a deep understanding of the four operations. For example, 3 + 4 looks completely different than 3 x 4. The shift from additive thinking to multiplicative thinking can be difficult for kiddos. Common Core makes the transition in 2nd grade as students use addition to find the total number of objects arranged in an array. In 3rd grade, students are expected to find whole-number products.

So, how can we help our kiddos develop strong mental images of addition and multiplication? Always present the concepts in the context of word problems, have students use manipulatives to act out the situations (concrete), and give them plenty of practice drawing models of the addition and multiplication situations (representational). Remember, models aren’t fancy art pictures…they’re simple models that focus on the math. Circles and squares can represent anything!

My I Have/Who Has game uses representations of both addition and multiplication. You can see it combines expressions (abstract) with models (representational). It’s a great whole class game, but you can also throw it in a workstation and just let the kids sequence the cards. The easiest way is to lay all the cards out face up (in an array…ha ha). Players start with any card, find the card that follows it, and the next one, etc. Make sure they are reading the cards out loud to bring in the communication component. If they do it right, they should end up with the card they started with.


  1. Fancy Nancy

    I love this I Have Who Has!!! I feel like you are my personal mentor. Your blog and products are so spot on for what I need to be a better teacher. (grade 3) Thank you.

    • Donna Boucher

      What a sweet, sweet comment! I’m so pleased you find my blog helpful. 🙂

  2. Bonnie Condit

    I am a third grade teacher, and I love your site! You have some amazing ideas; thank you for sharing! However, I recently purchased your Common Core: I Have/Who Has Addition and Multiplication Models. I tried this game with my class but we couldn’t make it through a complete cycle of the cards. I realized there are TWO cards that ask, “Who has 6 + 1?” Just thought you might want to know. 🙂

    • Donna Boucher

      Thank you so much for letting me know! It always amazes me when a product has been downloaded MANY times, and no one has let me know there’s a mistake. I’m uploading the corrected version now!


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