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Concrete-Representation-Abstract

  • Visuals Aid Memory

    This research has major implications for math for students with special needs…but some of us already knew this!

  • Hands on Triangles for Boring Worksheet

    Hands on Triangles for Boring Worksheet

      A student came to me with a geometry worksheet, excerpt in photo above. Extemporaneously I created cut out sides of a triangle to help make the concept of lengths of sides of a triangle more concrete. The concept is that the shorter 2 sides must be longer than the 3rd side or you cannot…

  • Function Notation

    Function Notation

    Meaningful way to present function notation to avoid common misconceptions, e.g. y(5) is “y times 5”

  • – symbol: negative vs subtraction

    – symbol: negative vs subtraction

    The use of the “-” symbol is challenging for many students. They don’t understand the difference between the use of the symbol in -3 vs 5 – 3. To address this I use a real life example of multiple uses of the same symbol (1st 2 photos below) then break down the “-” symbol (photo…

  • Fractions! Meaning Making for Comparing Fractions

    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…

  • Snow Math

    Here’s a common word problem used for linear functions and equations (y=mx+b): There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6. Often the…

  • Factoring Numbers

    Factoring, as used in elementary school,  is the act of changing a number into numbers that multiply to produce the original number. For example, 12 can be factored to 3×4. 3 and 4 are called factors of 12. In the photo above, a “factor tree” is used to help identify the factors. I have often…

  • FRACTIONS! Meaning Making for Adding Fractions

    Fractions is one of the most challenging topics in math. Here’s an approach to help introduce fractions. I show the photo above, explain to a student that he and I both paid for the pizza. We are going to finish eating the pizza and I get the slice on the left. I ask “is this…

  • Corresponding Angles in Stain Glass

    Found this (above) cool example of corresponding angles (see photo below for explanation). This window photo could be a nice introduction to this type of problem by printing it out on paper and having students match angles as the teacher shows the photo on the Smart Board or screen.

  • Introduction to Slope

    Introduction to Slope

    Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above).  This includes presenting the topic using multiple representations: the original real life situation, rates (see photo…