Fractions is one of the most challenging math topics. Many high school and college students struggle to some degree with fractions. The Common Core of State Standards (CCSS), despite all the criticism, includes components to address the conceptual understanding of fractions. Below is a photo showing a 4th grade Common Core standard regarding fractions along with an objective for a class lesson I taught at an elementary school in my district. I subsequently presented on this at the national CEC conference in 2014. Notice the bold font at the bottom, ¨justify…using a visual fraction model.¨ The photo above shows an example of a model I used in class.
The photo below shows a handout I used in the lesson. The first activity involved having students create a Lego representation of given fractions. These would eventually lead to the photo at the top with students comparing fractions using Legos. The students were to create the Lego model, draw a picture version of the model then show my co-teacher or I so we could sign off to indicate the student had created the Lego model.
The Lego model is the concrete representation in CRA. In this lesson I subsequently had students use fractions trips (on a handout) and then number lines – see photos below.
The following shows steps to introducing the concept of the value of money and of adding coins.
The concept of a dime is presented as 10 pennies (see below). The dime is compared to a penny, nickel and quarter using these representations. Repeated use of these representations leads into an intuitive understanding of the coins.
Next is determining the value of multiple coins. The place to start is with pennies, which is relatively easy as the number of pennies represents the value. The next step is to count dimes because counting by 10s is relatively easier than counting by 5s or 25s.
Dr. Russell Gersten is a guru in special ed. At a presentation at the 2013 national Council for Exceptional Children he explained that number sense is best developed using the number line. With this in mind I created a CRA approach using the number line.
First, the student lines up the dimes on the number line (see photo below) then skip counts to determine the cardinal value, which is the value of the coins.
Upon demonstrating mastery of counting dimes, the student moves from using coins (concrete) to a representation – see photo below.
This approach is used for nickels and then a combination of nickels and dimes (corresponding blog post forthcoming).
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.