Understanding Common Core Style Models
from elementary through high school, consistency is key
Common Core Math Standards have been under intense scrutiny often with individual problems being picked apart instead of examining the larger picture of learning over time.
I’ll take a set of 3rd grade CC standards and show you how teachers can build on these methods to increase understanding and create stronger math and problem solving skills.
Some Common Core Math Standards
The following is a subset of the Grade 3 Measurement and Data CC Standards. Let’s look at how these standards translate in the classroom and how teachers can utilize these methods.
Lower Elementary
Representing Area
For many students area models are not only real world applications but also essential visual guides. For example, the area of this rectangle is represented by the multiplication 10 x 25.
What happens to the area when I divide the rectangle into two?
The area doesn’t change, but how it is distributed does change. This is a geometric model of the distributive property.
Upper Elementary
Multiplying Double-digit Numbers
Teachers can extend the model to 2-by-2 multiplication as well. For example, let’s model 17 x 28 as the area of a 17 by 28 unit rectangle.
Section the rectangle by separating each side length into tens and ones.
Sum the four areas to find the total area.
You may be thinking this is overcomplicated.
Wouldn’t it be simpler to multiply 17 x 28 using regular multiplication? Why draw all these rectangles?
To understand the purpose of this exercise, one must understand its relationship to multiplying binomials.
A binomial is an expression with two terms. To demonstrate, I could write the above problem as the following product:
Multiply both terms in the first parentheses to both terms in the second to obtain the same four products as in diagram 5.
Algebra 1
Multiplying Binomials
Algebra is notorious for placing unknowns into common processes. So let’s replace 10 and 20 from the above problem with x and 2x, respectively.
Again the model represents the product of two binomials.
The combined area represents the polynomial solution after multiplying.
To get this answer algebraically students learn the FOIL method.
FOIL is a sound technique but doesn’t show why it works. By learning the area model students can see why the mnemonic is handy. Once they understand why it works, they then can use FOIL (or some equivalent method) as an algebraic shortcut.
Multiplying Trinomials
Using the model, students can multiply higher degree polynomials too.
If you only knew the FOIL method, multiplying trinomials would be a stretch since they’re not binomials. By using area models students can extend the diagram to discover the correct distributions on their own.
To do this draw lengths to represent the trinomials. Then draw the corresponding boxes and fill in the areas as we did before.
Lastly add together like terms to find the solution:
Once students understand the distributive process, they can leave behind the diagram and work purely with the algebraic methods.
Algebra 2
Completing the Square
Algebra 2 teachers can build on these ideas even further! In second year algebra students learn how to complete the square.
Generally, students are given the following instructions with an example:
Set the equation equal to zero. Subtract the constant. Take the middle term’s coefficient, divide it by two, square it and add it to both sides of the equation. Then factor the trinomial into a binomial squared and add the constant back over.
The instructions work perfectly, but are easy to forget when not fully understood. The geometric model fills this gap in understanding and helps students remember the algorithm.
Here’s how to complete the square of this equation using models.
Construct a large square with a smaller square inside to represent x-squared, consistent with our previous models. The middle term in the equation is 2x. Divide it evenly between the 2 light blue boxes.
I can then deduce that the lengths of the large square must be x and 1.
To complete the square, I need to find the area of the purple box. Since its side lengths are both 1, the area must be 1 x 1.
Of course I can’t just add 1 to my equation. Instead borrow 1 unit from the 12 unused units in the original equation to complete the square.
Lastly write the equation as the square of (x + 1) with 11 units extra.
College
The Euclidean Algorithm
Translating algorithms into area models is a powerful tool students can use beyond high school for problem solving and understanding.
For example, in a recent lesson I taught the Euclidean Algorithm, a common Number Theory topic. I also showed a geometric explanation of the algorithm using area models to increase understanding.
Next Lesson: Strategies for Mentally Adding Number Lists
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