Equivalent Fractions Using Area Models & Number Lines - Fourth Grade Mathematics
When two different names refer to the very same person—such as Robert and Bob—we know they are two expressions for the exact same human being. In the mathematical universe of fractions, an identical phenomenon occurs constantly. The fractions 1/2, 2/4, 4/8, and 50/100 look completely different on the surface because their numerators and denominators are different digits. Yet, every one of these fractions represents the exact same quantity of space and marks the exact same physical coordinate on a number line. Fractions that represent the same value are called equivalent fractions. In this chapter, you will use visual area models and precision number lines to prove fraction equivalence and discover how partitioning creates new names for identical amounts.
The Area Model Proof of Equivalence
An area model proves equivalence by demonstrating that the total shaded area remains completely unchanged when pieces are partitioned into smaller subdivisions.
+-----------------------------------------------------------------------------------+
| AREA MODEL PROOF: 1/2 IS EQUIVALENT TO 2/4 |
+-----------------------------------------------------------------------------------+
| |
| MODEL A: One Whole split vertically into 2 equal parts (Halves): |
| +--------------------------------+--------------------------------+ |
| | | | |
| | SHADED (1/2) | Unshaded | |
| | | | |
| +--------------------------------+--------------------------------+ |
| |
| MODEL B: Cut horizontally across to double the number of parts (Fourths): |
| +--------------------------------+--------------------------------+ |
| | SHADED (1/4) | Unshaded | |
| +--------------------------------+--------------------------------+ |
| | SHADED (1/4) | Unshaded | |
| +--------------------------------+--------------------------------+ |
| |
| Total Shaded Area in Model B: 1/4 + 1/4 = 2/4. |
| Notice: The shaded region did not grow or shrink by a single millimeter! |
| Therefore: 1/2 = 2/4. |
| |
+-----------------------------------------------------------------------------------+
The Slicing Action: Multiplying by One in Disguise
What happened when we drew that horizontal line across Model A? Every single piece was split into 2 smaller pieces. The number of shaded pieces doubled: 1 piece became 1 x 2 = 2 pieces. The total number of pieces in the whole doubled: 2 pieces became 2 x 2 = 4 pieces. Both the numerator and the denominator were multiplied by 2: (1 x 2) / (2 x 2) = 2/4. Because you multiplied the top and bottom by the same number, you multiplied by 2/2, which is equal to 1 whole! Multiplying any number by 1 leaves its value unchanged.
Number Line Equivalence: Sharing Coordinates
When fractions are equivalent, they occupy the exact same point on a number line between 0 and 1.
+-----------------------------------------------------------------------------------+
| EQUIVALENT FRACTIONS ON NUMBER LINES |
+-----------------------------------------------------------------------------------+
| |
| Thirds: |
| 0 1/3 2/3 3/3 = 1 |
| +---------------------+---------------------+---------------------+ |
| | |
| | (Exact same coordinate!) |
| Sixths: v |
| 0 1/6 2/6 3/6 4/6 5/6 6/6 = 1 |
| +----------+----------+----------+----------+----------+----------+ |
| |
| Conclusion: 2/3 and 4/6 land on the exact same vertical coordinate. |
| Therefore: 2/3 = 4/6. |
| |
+-----------------------------------------------------------------------------------+
Why Coordinate Alignment Proves Equality
A number line represents distance from 0. If a runner starts at 0 and runs 2/3 of a mile, and another runner starts at 0 and runs 4/6 of a mile, both runners are standing at the exact same physical spot on the path. Their distances from 0 are identical. This geometric alignment proves that 2/3 and 4/6 are mathematically equal.
Generating Equivalent Fractions Geometrically
You can generate an infinite family of equivalent fractions by continuing to slice area models.
+-----------------------------------------------------------------------------------+
| THE EQUIVALENT FRACTION FAMILY |
+-----------------------------------------------------------------------------------+
| |
| Base Fraction: 3/4 |
| |
| Slice each piece into 2 parts: (3 x 2) / (4 x 2) = 6/8 |
| Slice each piece into 3 parts: (3 x 3) / (4 x 3) = 9/12 |
| Slice each piece into 4 parts: (3 x 4) / (4 x 4) = 12/16 |
| Slice each piece into 10 parts: (3 x 10) / (4 x 10) = 30/40 |
| |
| All members of the family represent the exact same amount: |
| 3/4 = 6/8 = 9/12 = 12/16 = 30/40 |
| |
+-----------------------------------------------------------------------------------+
Chapter Practice Exercises
Exercise 1: Draw or describe an area model that proves 2/5 is equivalent to 4/10. Explain what slicing action occurred.
Exercise 2: Use number lines to show that 3/4 and 6/8 represent the same location. Describe the intervals between 0 and 1 for both lines.
Exercise 3: Generate three fractions equivalent to 2/3 by multiplying the numerator and denominator by 2, 3, and 5.
Exercise 4: Determine whether the fractions 3/6 and 5/10 are equivalent. Use both an area model description and multiplication/division reasoning to justify your answer.
Exercise 5: A student looks at an area model for 1/3 and cuts each of the 3 sections into 4 equal pieces. The student shades 4 of the new pieces. What fraction of the whole is shaded, and what equivalence did the student prove?
Solutions and Step-by-Step Explanations
Solution 1: To prove 2/5 = 4/10, start with a rectangle divided into 5 equal vertical columns, with 2 columns shaded to represent 2/5. Draw a horizontal line across the middle of the rectangle, cutting every piece in half. There are now 10 total equal pieces in the rectangle, and 4 of those pieces are shaded. Since the total shaded area did not change, 2/5 is equivalent to 4/10.
Solution 2: On the first number line, divide the distance from 0 to 1 into 4 equal fourths. The fraction 3/4 is located at the 3rd tick mark. On the second number line of identical length, divide the distance from 0 to 1 into 8 equal eighths. The fraction 6/8 is located at the 6th tick mark. Looking vertically, the 3rd tick mark on fourths aligns precisely above the 6th tick mark on eighths. Both fractions represent the exact same distance from 0, proving 3/4 = 6/8.
Solution 3: Multiplying numerator and denominator: Multiply by 2: (2 x 2) / (3 x 2) = 4/6. Multiply by 3: (2 x 3) / (3 x 3) = 6/9. Multiply by 5: (2 x 5) / (3 x 5) = 10/15. Three equivalent fractions are 4/6, 6/9, and 10/15.
Solution 4: Both 3/6 and 5/10 are equivalent. Using an area model, both fractions shade exactly half of the whole figure. Using arithmetic: dividing the numerator and denominator of 3/6 by 3 gives 1/2. Dividing the numerator and denominator of 5/10 by 5 gives 1/2. Because both fractions are equivalent to 1/2, they are equivalent to each other: 3/6 = 5/10.
Solution 5: When each of the 3 sections is cut into 4 equal pieces, the whole contains 3 x 4 = 12 equal pieces. Shading 4 of the new pieces represents the fraction 4/12. Since the 4 small pieces occupy the exact same space as the original 1 third, the student proved that 1/3 is equivalent to 4/12.