Place Value Representation

Here is an easy way to create and implement strategy to unpack place value for students (created by one of my former graduate students). I suggest using this after manipulatives and visual representations (drawing on paper) in a CRA sequence. It is hands on but it includes the symbolic representation (numbers). Hence is another step before jumping into the mathy stuff.

pace value representation just numbers.jpg

The focus can shift to money as well.

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Cutouts for Area of Irregular Shapes

Students have trouble with irregular shapes largely because they cannot visualize or determine the measures the dimensions of the individual parts. The photos below show a hands on activity to help students with these challenges. The students are cut out the individual shapes and write in the dimensions. This method allows them to see the individual parts and the respective dimensions. (The second has calculation errors.) This activity is followed by a handout in which students can shade in the different parts which is a step towards more abstraction – CRA.trapezoid area cut outstrapezoid area cut outs two triangles

Conceptual Presentation of Percents




Percents is a very challenging concept for students often because it is presented in symbolic terms, e.g. 40% of a 25 is what? Students typically understand what 100% and 50% mean. The concept of percent can build on this intuitive understanding. Here is my approach using a Concrete-Representation-Abstract approach (from a lesson taught today).

First, I ask the students to match common terms used for various percents on a percent number line. The students typically understand the first four terms.

percent number line words



I build on their knowledge of 50% first by having them cut (see below) a pizza into 50% then 25%. Then they cut $200 into 50% then 25% and then count the money.


percents cut outs


Many struggle with 1/4 and 3/4 (25% and 75%). The little less than half is the building block because it is the first step away from 0%, 50% and 100%. Here’s how I use this step (below). The students compute 50% or half of the $200 then estimate 40% which is “a little less than 50%.” If a student struggles with this step, we go back to the cutout money and they cut a little off of the 50% and guess how much money is left.

money on percent number line


One model for memory is called the Information Processing Model or Dual Storage Model.


Here’s the suggested process in this model in a class instruction context:

  1. Our senses receive stimuli. In the classroom students hear the teacher or a classmate talking, see the teacher’s notes or the note being passed to them, smell various things in class, taste their gum etc. 
  2. The sensory register filters out most stimuli which means the teacher’s lesson is competing with all the other stimuli for attention. Most students are either visual or hands on learners yet the majority of instruction is conducted through auditory means. Information in a lesson that is meaningful or interesting is more likely to make it through the register.
  3. The information that makes it to the working memory is processed. Working memory has a limited capacity. Like a computer, if it is attempting to process a lot at one time it slows down. It is hard for some students to process a lot of auditory information if they are a visual learner so as they are attempting to process they may be missing other parts of instruction. This is why scaffolding and other strategies are important. They help reduce the amount of information the student has to process. The working memory also attempts to organize and make sense of the information -Gestalt Theory. In the photo below are some examples. When I present the image below under “closure” and ask people what they see, the response is almost always “a triangle.” The really is no triangle there but the brain fills in the gaps. The brain wants to make the visual information meaningful. 
  4. The information that makes it to long term memory is filed away. Effective learning means the stored information can be readily retrieved. Think of computer files or files in a file cabinet. I have a file for Gabriel’s IEPs so I can easily retrieve them. Contrast that with how a student may stuff his homework assignment into his bookbag but later cannot find it. Effective storage is enhanced when the information is organized and makes sense. This is helped by making the information meaningful or by addressing prior knowledge (e.g. new IEPs filed with old IEPs).

Most if not all educational strategies would address some aspect of this model.

Representations of Concepts

Representations of Concepts

Three ways to represent perimeter: I taught a lesson on perimeter to a 5th grade class. First I had them create a rectangular pen for their animals and they counted the number of fence pieces. Then we drew a rectangle to represent the pen. Finally we looked at the formula. This allows a deeper conceptual understanding of the concept. This is known as Concrete-Representation-Abstract – representing the concept at all three levels.

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