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Students can create models that represent their understanding, but can they undo models that explain the reasoning of others?  It was time to find out.  I showed this picture to a 5th grade class and asked them to write an equation or expression that matched the area model… Every student in the class said it was modeling multiplication however no student could account for the 5 squares at the bottom.

Why did this happen?

My takeaways: In order for students to understand what’s happening, there’s a progression of understanding that begins all the way back in 2nd grade.  This understanding is built through exploration.  These explorations happen in and out of context, with and without manipulatives and representations, but most importantly…it happens over time.

2nd Grade: students are expected to partition a square in to 5 rows and 5 columns.  They should also use additive equations that match the model.

• 2.OA.4 – Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns

3rd Grade: students begin to model multiplication AND division with arrays at the same time to develop relational thinking.

• 3.OA.1 – Interpret products of whole numbers
• 3.OA.2 – Interpret whole-number quotients of whole numbers The model above is typically described as 3 x 6 but it could also represent division by giving students 18 tiles and have them construct a rectangle with 3 rows.  The 18 tiles now become the dividend (not the product), which takes flexibility in understanding.

What happens if you ask students to build a model with 19 tiles and place them in 3 rows?

19 can be divided into three rows but the context will determine what to do with the 1 remaining tile. The blue tile can be recognized as a remainder, as a third, or as 1/6 of the next group.  Using “untidy” number combinations can open the door to some awesome discussion.

Third grade students also begin to multiply with single-digit factors and multiples of 10.

• 3.NBT.3 – Multiply one-digit whole numbers by multiples of 10 in the range of 10-90 using strategies based on place value and properties of operations. If multiplication and division are taught concurrently, then students begin to understand that the model below has multiple meanings but it depends on the context.  Flexibility is really hard to teach.  So don’t teach it…provide opportunities that foster flexible thinking. The difficult thing for students to recognize ties back to an understanding that is built in kindergarten…unitizing.  Even though these are “ten rods”, students need to see them as a rod composed of ten ones.  If students only see them as rods of 10 it becomes extremely difficult to see this model as division.

4th Grade: students begin to explore multiplication and division with multi-digit numbers.  