Mathematics Institute for Learning Disabilities and Difficulties

MSTAR Interventions

Instructional Background

Preskills Knowledge and Skills for Students

  • Understand when a fraction can and cannot be used. Use a fraction to describe only a whole that is divided into equal parts.
  • Identify the numerator and denominator of a fraction when represented in fraction form, as a set, as an area model, or on a number line.
  • Interpret the numerator as the number of selected parts.
  • Interpret the denominator as the type of parts or the number of total parts.
  • Read and write fractions represented in fraction form, using correct mathematical language.
  • Use multiplication and division facts from 1 to 12.
  • Generate multiples and factors.

Mathematics Content Knowledge for Teachers

  • Fractions express numbers that are between whole numbers.
  • A fraction represents a single number or location on the number line.
  • Fractions are rational numbers.
  • Whole numbers are a subset of rational numbers.
  • Fractions define the relationship between parts and wholes (objects, groups of objects, or quantities).
  • A set is a collection of individual objects. Each object is an equal part of the whole.
  • A fraction always represents a part of something. This “something” is called the whole or 1.
  • The parts into which the whole are divided must be equal.
  • The number above the fraction bar is the numerator. It is the number of parts.
  • The number below the fraction bar is the denominator. It is the type of parts or the total number of parts that make up 1 whole.
  • An infinite number of different fractions can represent a single location on the number line. For example,  ,  ,  ,  , etc. represent a single location on the number line.
  • The fraction bar is also called a “vinculum.” A vinculum is a horizontal line to indicate that multiple terms are considered as a single term.
  • Unit fractions are formed by dividing a whole (region, set, or distance on a number line) into equal parts1.
  • If the whole is divided into B equal parts, then the amount formed by 1 of those parts is  of the whole. In other words, B copies of the amount  of the whole are joined together make the whole.  is the unit fraction2.
  • 1 whole consists of B pieces of size .
  • Where A and B are counting numbers, the fraction  of the whole is the amount formed by A parts (or copies of parts), each of which is  of the whole2.
  • Using the English fraction term,  is read: “four-sixths.” The number of parts being described is said first, “four.” Then, the name of the parts in the whole is said, “sixths.” 6 parts are involved, so each part is  of the whole1.
  •  can also be described as “out of 6 parts, select 4.” The number of the parts in the whole is indicated first, and then the number of parts is stated1.
  • The multiplicative identity property, also called the identity property of 1 or identity property of multiplication, states that the product of any number and 1 is the number itself.
  • 1 is the multiplicative identity. 1 can be represented as  ,  ,  , etc.
  • Because whole numbers are a subset of rational numbers, which include fractions, the multiplicative identify property applies when the original number is a fraction and when 1 is written in fractional form.
  1. National Council of Teachers of Mathematics. (2009). Focus in grade 3: Teaching with curriculum focal points. Reston, VA: Author.
  2. Beckman, S. (2011). Mathematics for elementary teachers with activity manual (3rd ed.). Boston, MA: Addison-Wesley.