Mathematics and Science Institute for Students With Special Needs

MSTAR Interventions

• Equivalent Fractions
   
• Ratios and Rates
   
• Facts & Patterns:
  Multiplication & Division
• Proportionality
   

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.