Ch. 12: Exploring Fraction Concepts (Grades 3-5) (Fractional Parts of a…
Ch. 12: Exploring Fraction Concepts (Grades 3-5)
Meanings of Fractions
"Sometimes the words whole and unit are used interchangeably in discussing fractions, but the unit must be thought abut to interpret all the possible concepts."
Effective starting point for building meaning of fractions.
3/5 of class went on field trip.
Identifying an amount of a continuous unit (length, area, volume, or time) (Ex. 1/2 of a mile).
Equal shares: Ex. Share $10 equally among 4 people.
Fractions can be used to indicate an operation (Ex. 4/5 of 20 square feet).
Ratio (Gr. 6)
Expresses a relationship between two quantities or parts of quantities and compares their relative measures or counts (Ex. 3/4 of the class is wearing jackets, part, and the other 1/4 isn't wearing jackets, part, to the whole class, whole).
Common Errors or Misconceptions
Numerator and denominator are separate numbers.
Fractional parts do not need to be equal-sized.
Fractional parts must be same shape.
Fractions with larger denominators are bigger.
Fractions are subtraction.
Fractions with larger denominators are smaller.
Models for Fractions
"Effectively using physical models in fraction tasks is important."
Area: Fractions are determined based on how a part of a region or area relates to the whole area or region.
Length: Fractions are represented as a subdivision of a length of a paper strip (representing a whole), or as a length/distance between 0 and a point on a number line, subdivided in relation to a given whole unit.
Set: Fractions are determined based on how many discrete items are in the whole set, and how many items are in the part.
Fractional Parts of a Whole
Fraction size is relative
"A key idea about fractions is that a fraction does not say anything about the size of a whole or the size of parts. A fraction only tells us about the relationship between the part and the whole."
"Do you want half of this pizza or one third of that pizza?"
It depends on how big each of the whole pizzas are.
Partitioning with area models
Students need to be aware that 1) the fractional parts must be the same size, though not necessarily the same shape and 2) the number of equal-sized parts that can be partitioned within the unit determines the fractional amount."
Partitioning with length models
Number lines can be challenging because students may ignore the size of the interval (Ex. Activity 12.5 Pg. 229).
Partitioning with set models
Coins, counters, baseball cards, etc. (Ex. Figure 12.6 Pg. 230)
Counting fractional parts, iterating, helps students understand the relationship between the parts (the numerator) and the whole (denominator) (Ex. Figure 12.9 Pg. 231).
"The way that we write fractions with a top component and a bottom component and a bar between is a convention, an agreement for how to represent fractions."
What does the denominator/numerator in a fraction tell us?
What might a fraction equal to 1 look like?
Magnitude of Fractions
Students should be able to tell how big a particular fraction is and should be able to tell which of two fractions is larger or smaller.
Conceptual focus on equivalence
Ex. How do you know that 4/6=2/3?
They are the same because you can simplify 4/6 and get 2/3. If you have a set of 6 items and you take 4 of them, that would be 4/6. But you can put the 6 items into 3 groups, and the 4 items would then be 2 groups of the 3 groups. That means it's also 2/3.
Equivalent fraction models
"Starting with a context and physical models is a perfect starting point for helping students create an understanding of equivalent fractions." (Ex. Figure 12.14 Pg. 241).
Developing an equivalent-fraction algorithm
"When students understand that fraction can have different (but equivalent) names, they are ready to develop a method for finding equivalent names for a particular value. (Gr. Late 3rd-5th) (Pg. 245).
Using number sense
Comparing unit fractions
Understanding that just because 12 is larger than 3, does not mean that twelfths is more than thirds."
Comparing any fractions
"Goal is for students to select efficient strategies for determining the larger fraction, not memorizing an algorithmic method of choosing the correct answer."
Using equivalent fractions
Equivalent-fraction concepts can be used in making comparisons. (Ex. Which of these fractions is greater or are the fractions equal?) ("This question leaves open the possibility that two fractions that may appear different, can in fact, be equal).
Teaching Considerations for Fraction Concepts
Give a greater emphasis to number sense and the meaning of fractions, rather than rote procedures for manipulating them.
Provide a variety of physical models and contexts to represent fractions.
Emphasize that fractions are numbers, making extensive use of number lines in representing fractions.
Spend whatever time is needed for students to understand equivalences (concretely and symbolically), including flexible naming of fractions.
Link fractions to key benchmarks and encourage estimation.