Graphic better than FOIL

Typically, multiplying two binomials is presented with FOIL. This approach is problematic for two reasons: it is a mneumonic for a purely symbolic representation and it is also an isolated strategy that does not connect well to prior knowledge.


Another approach is to use a graphic organizer and context. In the photo below I presented students with the scenario of expanding my patio (under the guise of being so popular I needed more space to entertain). The top figure shows the expansion of length alone. This allows for a simple distributive problem and (and eventually factoring out the GCF). The bottom figure shows an expansion of length and width which leads to multiplying binomials. Students would see that the area of the new patio is computed with (6 + x)(5 + x) which is the same as finding the area of each individual rectangle (presented in photo at the bottom). (This is useful for factoring trinomials as well.)


This scaffolded approach is effective because it presents the concept in a different representation, it connects to prior knowledge of distributive property (useful for memory storage) and it is connected to prior knowledge of area of rectangles so it has meaning.

Subsequent problems would use the boxes as a graphic organizer (see photo below).

multiply binomials

Color Coding and Different Representation for Rounding

color coding for rounding adjusted (2)


There are two strategies used in this example for rounding.

The number line is a different representation for a rounding situation. In CRA this is the representational or pictorial level. Typically students are taught to round by looking only at the numbers which is purely symbolic.

The color coding helps the students discriminate between the number being rounded and the choices for rounding. As I’ve written previously, color coding helps a student discern different parts of a concept.


The handouts are found at Super Teacher Worksheets.

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