Fraction mathematics models are visual tools that help make abstract fraction concepts more concrete. The measurement, partitive, and rectangular area models assist students in understanding what fractions represent, how they work, and how they relate to each other. Although these three models are different, they often overlap. For instance, a number line can be seen as a stretched area model, and an area model can be used to solve partitive problems.
Summary Table of Fraction Models
| Model | Primary Focus | Best Used For |
|---|---|---|
| Measurement | Length/Position | Ordering, Comparing, Number Line |
| Partitive | Sharing/Size of Groups | Unit Fractions, Division as Sharing |
| Area | Surface/Shading | Multiplication, Equivalence, Addition |
Measurement Model (Linear/Number Line)
The measurement model (or length model) defines fractions as a specific distance or length from zero on a number line.
- Key Concept: Fractions are numbers with a specific position relative to the unit (1 whole).
- Usage: It is best for comparing fractions, ordering them, and understanding their size.
- Fraction Measurement Models: Support understanding various aspects of fractions, including their size, equivalence, and comparison. A fraction measurement model can be understood through various representations including:
- Linear Models: These use lengths to represent fractions, allowing students to visualize how fractions relate to whole numbers. To represent 1/4 on a number line, you divide the interval between 0 and 1 into four equal segments and mark the point at the end of the third segment, e.g., on a tape measure.
- Area Models: Fractions can be represented as areas of shapes, helping to understand the size of fractions visually.
- Set Models: These models show fractions as parts of a whole, which can be useful for comparing fractions.
- Visual Fraction Models: Tools like pie charts and bar models help visualize fractions and their relationships, making abstract concepts more concrete.
- Linear Models: These use lengths to represent fractions, allowing students to visualize how fractions relate to whole numbers. To represent 1/4 on a number line, you divide the interval between 0 and 1 into four equal segments and mark the point at the end of the third segment, e.g., on a tape measure.
Partitive Model (Sharing/Set Model)
Partitive division, also known as sharing or grouping division, is a method of dividing a total quantity into a specified number of groups. In this model, the focus is on determining how many items will be in each group when the total is divided evenly among the groups.
- Key Concept: “How much is one of the parts?” The number of groups (denominator) is known, and we are finding the size of one part.
- Usage: Understanding unit fractions, sharing problems, and understanding the numerator.
- Partitive Models: Two ways of thinking about division, each with its own approach to understanding the concept. Both partitive and quotative division yield the same numerical answer but represent different aspects of the division problem. Understanding these models is crucial for developing a well-rounded understanding of division and its applications in mathematics. Here are the key differences between the two:
- Partitive Division: Partitive division, also known as sharing or grouping division, is a type of division problem where you know the total number of items and the exact number of groups you want to divide them into. The goal is to determine how many items will be in each group. For example, if you have 20 oranges and want to share them equally among 4 groups, partitive division helps you find that each group receives 5 oranges. For example, dividing 12 cookies among 4 children results in each child getting 3 cookies.
- Measurement (Quotative) Division: Quotative division, also called measurement division, involves splitting a number or set of items into groups of a predetermined size, with the goal of finding the number of complete groups that can be made from the total quantity. Unlike partitive division, where the total number of groups is known and we find the size of each group, in quotative division the size of each group is known, and the unknown is the number of groups. A student has 48 toffees and distributes them among classmates, giving each 4 toffees. To find the number of students, we calculate 48 ÷ 4 = 12 students.
Rectangular Area Model (Region Model)
The rectangular area model is a visual tool used to represent fractions as rectangles. This model allows students to see the fraction as a shape divided into equal parts, making it easier to understand and compare fractions. This model can also provide a clear visual representation of the multiplication of fractions, helping students understand the concept without the need for complex calculations.
- Key Concept: The total shaded area relative to the total area is the fraction.
- Usage: The rectangular area model is the most common model for teaching fraction multiplication, equivalence, and addition/subtraction with unlike denominators.
- Example Multiplication of Fractions: Multiply 2/5 × 3/4 = 6/20
- Representing the First Fraction: The denominator of the first fraction indicates the number of columns to divide the rectangle into, while the numerator tells us how many of those columns should be shaded. For example, for the fraction 2/5, the rectangle is divided into 5 equal parts.
- Representing the Second Fraction: The denominator of the second fraction determines the number of rows to divide the rectangle into, while the numerator specifies how many of those rows should be shaded.
- Finding the Product: The numerator of the product is the number of sections that have been shaded twice (the overlapping area), while the denominator is the total number of sections within the rectangle. For 2/5 and 3/4, the product is 6/20.
- Representing the First Fraction: The denominator of the first fraction indicates the number of columns to divide the rectangle into, while the numerator tells us how many of those columns should be shaded. For example, for the fraction 2/5, the rectangle is divided into 5 equal parts.
- Example Equivalent Fractions:
- Example Addition of Fractions: Add 2/3 + 4/5 = 22/15
- Representing the Fractions: The first step when given an addition problem with unlike denominators would to be to draw the fractions using area models.
- Find the Least Common Denominator: List out the multiples of each denominator until we find something in common. The least common denominator will be the easiest to work with (especially when drawing). In this example, 15 is the least common denominator.
- Finding the Sum: Since 15 was our common denominator, we need to now make 15 pieces in each. This is easiest when making the original drawing with vertical lines, and then use horizontal lines at this step. The most important part of this is to understand that you are not changing the amount that you originally started with. You are just cutting their original into smaller pieces. This is why having a strong understanding of equivalent fractions is so crucial.
- Calculate the Sum: Once you have 15 pieces in each, use the drawings to find the new equivalent fractions. Count how many are colored in (numerator) and the total number of pieces (denominator) and add them together. You can turn the fraction into a mixed number, if necessary.
- Representing the Fractions: The first step when given an addition problem with unlike denominators would to be to draw the fractions using area models.
- Example Subtraction of Fractions: Subtract 4/5 − 2/3
- Representing the Fractions: As shown in the steps above when adding fractions.
- Find the Least Common Denominator: Perform this step the same as above when adding fractions.
- Finding the Difference: Since 15 was our common denominator, we need to now make 15 pieces in each.
- Calculate the Difference: Once you have 15 pieces in each, use the drawings to find the new equivalent fractions. Color in, or cross out, the number of squares in 4/5 (i.e., 15) that are colored in the the 2/3 square (i.e., 10), and the total number of pieces remaining with the same color, or not crossed out, is the difference. In our example, 2 green squares remain after coloring in 10 of them. Therefore, 4/5 − 2/3 = 2/15
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