Prior to my trip to Singapore to work with American-sponsored overseas schools I shared three goals that I had set for the training in my Common Core Connections a Continent Away post. One of those goals was that the teachers would leave the training with a greater understanding of effective teaching strategies and activities that help ALL students learn and understand mathematics.

There are a number of well-known instructional strategies and plethora of activities to choose from that serve as effective approaches in helping students learn. In the training I led in Singapore, I wanted to draw attention to the small things teachers do that can sometimes have the greatest impact on student understanding and learning. To drive the point home, I asked each of the participating teachers to draw a right angle prior to sharing the following classroom experience of mine from almost 20 years ago.

“It was a typical day in my 6^{th} grade math class. I was sailing through the day teaching a geometry lesson in which I was discussing right angles. There I stood at the overhead, drawing right angles like the one shown below in Figure 1, moving right along with the discussion when one of my students raised his hand and said, ‘Ms. Bryant, are there left angles?’”

**Figure 1**

As you might guess, there were a few giggles from classmates. But then I looked at the model I had drawn on the overhead and knew perfectly well where he was coming from. He was seeing what I was drawing but clearly wasn’t getting it. What was clear to me was that he was classifying the angle as a right angle based upon the fact that the angle opened to the right rather than considering the angle measure. I immediately starting rotating and flipping the overhead transparency to illustrate that no matter the orientation (Figure 2) of the angle, it was still a right angle because right angles are defined by their measure and not by the direction.”

**Figure 2**

I’ve used this story many times and it’s interesting that when I have teachers draw an angle before sharing the experience, the overwhelming majority of teachers draw a right angle that is oriented facing the right as in Figure 1. Fortunately, this was a relatively simple misunderstanding to clarify with such a little act. Simply rotating and flipping the angle on the overhead transparency proved to be a very easy and powerful way to clarify the misunderstanding. Using this simple visual representation really helped define and clarify a right angle. And you can bet that after this incident, when talking about right angles, I always include the flipping and rotating of the angle at the get-go.

The *Common Core State Standards in Mathematics* (*CCSSM*) Standards for Mathematical Practice stress the relevance and importance of students using a variety of symbols, models, and figures to abstract the mathematical elements of a situation or to help interpret a mathematical situation. The example problem below is aligned to *CCSSM* 6.RP.A.3c . This standard requires students to solve problems involving finding the whole, given a part and the percent . The guided steps in solving the problem include models to illustrate the mathematical situation and provide hints for finding a solution.

**Jim ate 2 slices of pizza, which is 25% of the pizza. How many slices in the whole pizza?**

**Figure 3**

**Figure 4**

Research has shown for special education students, instruction that involves extensive use of visual representations appears to be crucial, Gerstung and Clarke (2007, p. 2). Visual models and representations are also crucial components of programs used in nations that perform well on international comparisons, such as Singapore, Korea, or the Netherlands. LearnBop authors include a variety of models in our online tutorials to help students interpret mathematical situations and solve problems correctly.

So what are you waiting for? Starting today, with our free class or school trial, you can provide your students more opportunities to utilize various models and representations proven to be effective for solving problems aligned to the *CCSSM*!

We value your feedback so we invite you to share your questions, comments, and suggestions about LearnBop or ways you’re using LearnBop to support students in learning mathematics. Thanks!

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