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As far as visuals for multiplication and division, we use four models. Equal groups can be done using felt mats and gathering equal amounts of cubes, tiles, bears or other manipulatives onto each mat. Arrays can be made using manipulatives organized into rows and columns. Area models are like arrays (rows/columns) but include squares inside (color tiles make a good manipulative and grid/graph paper for drawing pictorial models). We also use number lines and double number lines. A student can "hop" to show the problem. For example, to show 3x4=12, hop from 0 to 4, then 8, then 12 to make 3 hops of 4. Underneath it, the student could draw hops for the commutative version, ie 4 hops of 3, and show that both equal 12. Lastly, students can draw bar models to represent a quantity, then repeat those bars horizontally to show the product. Division is modeled in all the same ways except you are starting with the whole amount/quotient, and working toward the parts (how many groups and how many in each group).
This graphic shows it well: https://nw5thgrade.wordpress.com/2015/08/29/ways-to-model-multiplication/
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